?Calculus Kernels The transformations in 12-15 should be familiar from calculus

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(T: R^{2} \rightarrow R^{2}, \quad T(x, y)=(-x, y)\) Equation Transcription: Text Transcription: T:R^2 right arrow R^2, T(x, y)=(-x, y)

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(T: R^{3} \rightarrow R^{2}, \quad T(x, y, z)=(2 x+3 y-z,-x+4 y+6 z)\) Equation Transcription: Text Transcription: T: R^3 right arrow R^2 , T(x, y, z) = (2x+3y-z, -x+4y+6z)

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(T: R^{3} \rightarrow R^{3}, \quad T(x, y, z)=(x, y, 0)\) Equation Transcription: Text Transcription: T: R^3 right arrow R^3 , T(x, y, z)=(x, y, 0)

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(T: R^{3} \rightarrow R^{3}, \quad T(x, y, z)=(x,-z, x-2 y, y-z)\) Equation Transcription: Text Transcription: T: R^3 right arrow R^3 , T(x, y, z)=(x, -z, x-2y, y-z)

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(D: C^{1} \rightarrow C, \quad D(f)=f^{\prime}\) Equation Transcription: Text Transcription: D:C^1 right arrow C, D(f)=f’

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(D^{2}: C^{2} \rightarrow C, \quad D^{2}(f)=f^{\prime \prime}\) Equation Transcription: Text Transcription: D^2 : C^2 right arrow C, D^2 (f) = f’’

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(L_{1}: C^{1} \rightarrow C, \quad L_{1}(y)=y^{\prime} p(t) y\) Equation Transcription: Text Transcription: L_1 : C^1 right arrow C, L_1 (y) = y’+p(t)y

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(L_{n}: C^{n} \rightarrow C, \quad L_{n}(y)=y^{(n)}+a_{1}(t) y^{(n-1)}+\ldots+a_{n-1}(t) y^{\prime} a_{n}(t) y\) Equation Transcription: Text Transcription: L_n : C^n right arrow C, L_n (y) = y^(n) +a_1 (t) y^(n-1) +... +a_n-1 (t) y’+a_n (t) y

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(T: M_{23} \rightarrow M_{32}, \quad T(A)=A^{T}\) Equation Transcription: Text Transcription: T: M_23 right arrow M_32 , T(A) = A^T

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(T: M_{33} \rightarrow M_{33},\) \(T=\left[\begin{array}{lcc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]=\left[\begin{array}{lll} a & 0 & 0 \\ 0 & e & 0 \\ 0 & 0 & i \end{array}\right]\) Equation Transcription: Text Transcription: T: M_33 right arrow M_33 , T[a b c _ d e f _ g h i]=[a 0 0 _ 0 e 0 _ 0 0 i]

Finding Kernels Finds the kernel for the linear transformations in Problems 1-11. Describe the kernel. \(T: P_{2} \rightarrow P_{3}, \quad T(p)=\int_{0}^{x} p(t)\ d t\) for fixed x Equation Transcription: Text Transcription: T: P_2 right arrow P_3 , T(p)= integral_0 ^x p(t) dt

Calculus Kernels The transformations in Problems 12-15 should be familiar from calculus. Identify each transformation and give its kernel. (Problem 14 can have many correct answers.) \(T: P_{2} \rightarrow P_{2}, \quad T\left(a t^{2}+b t+c\right)=2 a t+b\) Equation Transcription: Text Transcription: T:P_2 right arrow P_2 , T(at^2 +bt+c)=2at+b

Calculus Kernels The transformations in Problems 12-15 should be familiar from calculus. Identify each transformation and give its kernel. (Problem 14 can have many correct answers.) \(T: P_{2} \rightarrow P_{2}, \quad T\left(a t^{2}+b t+c\right)=2 a\) Equation Transcription: Text Transcription: T: P_2 right arrow P_2 , T(at^2 +bt+c)=2a

Calculus Kernels The transformations in Problems 12-15 should be familiar from calculus. Identify each transformation and give its kernel. (Problem 14 can have many correct answers.) \(T: P_{2} \rightarrow P_{2}, \quad T\left(a t^{2}+b t+c\right)=0\) Equation Transcription: Text Transcription: T:P_2 right arrow P_2 , T(at^2 +bt + c)=0

Calculus Kernels The transformations in Problems 12-15 should be familiar from calculus. Identify each transformation and give its kernel. (Problem 14 can have many correct answers.) \(T: P_{3} \rightarrow P_{3}, \quad T\left(a t^{3}+b t^{2}+c t+d\right)=6 a t+2 b\) Equation Transcription: Text Transcription: T: P_3 right arrow P_3 , T(at^3 + bt^2 +ct + d)=6at+2b

Superposition Principle For Problems 16-20, suppose that T: V ? W is a linear transformation from vector space V to vector space W. Also suppose that \(\bar{u}_{1}\) is a solution of \(T(\bar{u})=\bar{b}_{1}\), and that bar \(\bar{u}_{2}\) is a solution of \(T(\bar{u})=\bar{b}_{2}\). Then \(\bar{u}_{1}+\bar{u}_{2}\) is a solution of \(T(\bar{u})=\bar{b}_{1}+\bar{b}_{2}\); this is called the Superposition Principle, as first introduced in Sec 2.1. Use linearity to prove the Superposition Principle. Equation Transcription: Text Transcription: bar u_1 T(bar u) = bar b_1 bar u_2 T(bar u) = bar b_2 bar u_1 + bar u_2 T(bar u) = bar b_1 + bar b_2

Superposition Principle For Problems 16-20, suppose that T: V ? W is a linear transformation from vector space V to vector space W. Also suppose that \(\bar{u}_{1}\) is a solution of \(T(\bar{u})=\bar{b}_{1}\), and that bar \(\bar{u}_{2}\) is a solution of \(T(\bar{u})=\bar{b}_{2}\). Then \(\bar{u}_{1}+\bar{u}_{2}\) is a solution of \(T(\bar{u})=\bar{b}_{1}+\bar{b}_{2}\); this is called the Superposition Principle, as first introduced in Sec 2.1. Show that y=cos t - sin t is a solution of the nonhomogeneous linear equation y’’-y’-2y = 4 sin t - 2 cos t. Equation Transcription: Text Transcription: bar u_1 T(bar u) = bar b_1 bar u_2 T(bar u) = bar b_2 bar u_1 + bar u_2 T(bar u) = bar b_1 + bar b_2

Superposition Principle For Problems 16-20, suppose that T: V ? W is a linear transformation from vector space V to vector space W. Also suppose that \(\bar{u}_{1}\) is a solution of \(T(\bar{u})=\bar{b}_{1}\), and that bar \(\bar{u}_{2}\) is a solution of \(T(\bar{u})=\bar{b}_{2}\). Then \(\bar{u}_{1}+\bar{u}_{2}\) is a solution of \(T(\bar{u})=\bar{b}_{1}+\bar{b}_{2}\); this is called the Superposition Principle, as first introduced in Sec 2.1. Show that y = t2 - 2 is a solution of y’’ - y’ - 2y = 6 - 2t - 2t2. Equation Transcription: Text Transcription: bar u_1 T(bar u) = bar b_1 bar u_2 T(bar u) = bar b_2 bar u_1 + bar u_2 T(bar u) = bar b_1 + bar b_2

Superposition Principle For Problems 16-20, suppose that T: V ? W is a linear transformation from vector space V to vector space W. Also suppose that \(\bar{u}_{1}\) is a solution of \(T(\bar{u})=\bar{b}_{1}\), and that bar \(\bar{u}_{2}\) is a solution of \(T(\bar{u})=\bar{b}_{2}\). Then \(\bar{u}_{1}+\bar{u}_{2}\) is a solution of \(T(\bar{u})=\bar{b}_{1}+\bar{b}_{2}\); this is called the Superposition Principle, as first introduced in Sec 2.1. Use Problems 17 and 18 and the Superposition Principle to write the general solution of y’’ - y’ - 2y = 4 sin t - 2 cos t + 6 - 2t - 2t2. Equation Transcription: Text Transcription: bar u_1 T(bar u) = bar b_1 bar u_2 T(bar u) = bar b_2 bar u_1 + bar u_2 T(bar u) = bar b_1 + bar b_2

Superposition Principle For Problems 16-20, suppose that T: V ? W is a linear transformation from vector space V to vector space W. Also suppose that \(\bar{u}_{1}\) is a solution of \(T(\bar{u})=\bar{b}_{1}\), and that bar \(\bar{u}_{2}\) is a solution of \(T(\bar{u})=\bar{b}_{2}\). Then \(\bar{u}_{1}+\bar{u}_{2}\) is a solution of \(T(\bar{u})=\bar{b}_{1}+\bar{b}_{2}\); this is called the Superposition Principle, as first introduced in Sec 2.1. Generalize the Superposition Principle to three or more terms. Equation Transcription: Text Transcription: bar u_1 T(bar u) = bar b_1 bar u_2 T(bar u) = bar b_2 bar u_1 + bar u_2 T(bar u) = bar b_1 + bar b_2

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [0 0 _ 0 0]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 0 _ 0 -1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 0 _ 0 0]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{ll} 1 & 2 \\ 4 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 2 _ 4 1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{ll} 1 & 2 \\ 2 & 4 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 2 _ 2 4]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{ll} 1 & 1 \\ 4 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 1 _ 4 1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 1 1 _ 1 2 1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{lll} 1 & 2 & 1 \\ 2 & 4 & 2 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 2 1 _ 2 4 2]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{lll} 1 & 2 & 1 \\ 2 & 1 & 2 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 2 1 _ 2 1 2]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{lll} 1 & 3 & 1 \\ 2 & 2 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 3 1 _ 2 2 1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{ll} 1 & 1 \\ 1 & 2 \\ 1 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 1 _ 1 2 _ 1 1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{ll} 1 & 2 \\ 2 & 4 \\ 1 & 2 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 2 _ 2 4 _ 1 2]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [0 0 _ 0 0 _ 0 0]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{ll} 1 & 1 \\ 2 & 1 \\ 3 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 1 _ 2 1 _ 3 1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{lll} 1 & 2 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 2 1 _ 0 1 1 _ 0 0 1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 2 & 3 & 2 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 1 1 _ 1 2 1 _ 2 3 2]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{lll} 1 & 2 & 1 \\ 2 & 4 & 1 \\ 1 & 1 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 2 1 _ 2 4 1 _ 1 1 1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{lll} 1 & 2 & 1 \\ 3 & 2 & 2 \\ 2 & 3 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 2 1 _ 3 2 2 _ 2 3 1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{lll} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 2 0 _ 0 1 1 _ 0 0 1]

Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right]\) Equation Transcription: Text Transcription: T(bar v) = A bar v, T: R^n right arrow R^m [1 1 0 _ 0 1 0 _ 0 0 0]

Let T: Rn ? Rm be linear transformation, and let \(\left(\bar{v}_{1}, \bar{v}_{2}, \bar{v}_{3}\right)\) be a linearly dependent set in Rn. Prove that the set \(\left(T\left(\bar{v}_{1}\right), T\left(\bar{v}_{2}\right), T\left(\bar{v}_{3}\right)\right.\) is linearly dependent in Rm. Equation Transcription: Text Transcription: (bar v_1 , bar v_2 , bar v_3 ) (T(bar v_1 ), T(bar v_2 ), T(bar v_3 )

Let T: Rn ? Rm be a linear transformation, and let \(\left(\bar{v}_{1}, \bar{v}_{2}, \bar{v}_{3}\right)\) be a linearly independent set in Rn. Give a counterexample to show that \(\left(T\left(\bar{v}_{1}\right), T\left(\bar{v}_{2}\right), T\left(\bar{v}_{3}\right)\right.\) need not be linearly independent in Rm. Equation Transcription: Text Transcription: (bar v_1 , bar v_2 , bar v_3 ) (T(bar v_1 ), T(bar v_2 ), T(bar v_3 ))

Let T: Rn ? Rm be an injective linear transformation, and let \(\left(\bar{v}_{1}, \bar{v}_{2}, \bar{v}_{3}\right)\) be a linearly independent set in Rn. Prove that \(\left(T\left(\bar{v}_{1}\right), T\left(\bar{v}_{2}\right), T\left(\bar{v}_{3}\right)\right)\) must be a linearly independent set in Rm. Equation Transcription: Text Transcription: (bar v_1 , bar v_2 , bar v_3 ) (T(bar v_1 ), T(bar v_2 ), T(bar v_3 ))

Provide that if a linear transformation T maps two linearly independent vectors onto a linearly dependent set, then the equation \(T(\bar{x})=\overline{0}\) has a nontrivial solution. Equation Transcription: Text Transcription: T(bar x) = bar 0

Consider the transformation T: P2 ? R2 defined by \(T(p(t))=\left[\begin{array}{l} p(0) \\ p(1) \end{array}\right]\) For example, if p(t) = t2 -6t + 4, then \(T(p(t))=\left[\begin{array}{r} 4 \\ -1 \end{array}\right]\) (a) Prove that T is a linear transformation. (b) Find a basis for the kernel of T. (c) Find a basis for the image of T. Equation Transcription: Text Transcription: T(p(t)) = [p(0) _ p(1)] T(p(t)) =[4 _ -1]

Kernel and Images: Find the kernel and image of each linear transformation in Problems 46-51. \(T: M_{22} \rightarrow M_{22}, \quad T(A)=A^{T}\) Equation Transcription: Text Transcription: T: M_22 right arrow M_22 , T(A) = A^T

Kernel and Images: Find the kernel and image of each linear transformation in Problems 46-51. \(T: P_{3} \rightarrow P_{3}, \quad T(p)=p^{\prime}\) Equation Transcription: Text Transcription: T: P_3 right arrow P_3 , T(p)=p’

Kernel and Images: Find the kernel and image of each linear transformation in Problems 46-51. \(T: M_{22} \rightarrow M_{22}, \quad T\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{ll} a & b \\ b & c \end{array}\right]\) Equation Transcription: Text Transcription: T:M_22 right arrow M_22 , T[a b _ c d] = [a b _ b c]

Kernel and Images: Find the kernel and image of each linear transformation in Problems 46-51. \(T: M_{22} \rightarrow R^{2}\), \(T\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{l} a+b \\ c+d \end{array}\right]\) Equation Transcription: Text Transcription: T: M_22 right arrow R^2 , T[a b _ c d] = [a+b _ c+d]

Kernel and Images: Find the kernel and image of each linear transformation in Problems 46-51. \(T: R^{5} \rightarrow R^{5}, \quad T(a, b, c, d, e)=(a, 0, c, 0, e)\) Equation Transcription: Text Transcription: T: R^5 right arrow R^5 , T(a, b, c, d, e) = (a, 0, c, 0, e)

Kernel and Images: Find the kernel and image of each linear transformation in Problems 46-51. \(T: R^{2} \rightarrow R^{3}, \quad T(x, y)=(x+y, 0, x-y)\) Equation Transcription: Text Transcription: T: R^2 right arrow R^3 , T(x, y) = (x+y, 0, x-y)

Examples of Matrices Give examples of matrices A in M33 such that \(T(\bar{x})=A \bar{x}\) has the properties described in Problems 52-54. The Im(T) is the plane 2x-3y+z=0 Equation Transcription: Text Transcription: T(bar x) = A bar x

Examples of Matrices Give examples of matrices A in M33 such that \(T(\bar{x})=A \bar{x}\) has the properties described in Problems 52-54. The Im(T) is the line spanned by \(\left\{\left[\begin{array}{l} 2 \\ 0 \\ 0 \end{array}\right]\right\}\) Equation Transcription: Text Transcription: T(bar x) = A bar x {[2 _ 0 _ 0]}

Examples of Matrices Give examples of matrices A in M33 such that \(T(\bar{x})=A \bar{x}\) has the properties described in Problems 52-54. The Ker(T) is spanned by \(\left\{\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \\ 2 \end{array}\right]\right\}\) Equation Transcription: Text Transcription: T(bar x) = A bar x {[1 _ 0 _ 1], [0 _ 1 _ 2]}

True/False Questions Answer Problems 55-60 true or false, and give a brief explanation or counterexample. If A is a square matrix, then \(\operatorname{Ker}\left(A^{2}\right)=\operatorname{Ker}(A)\). True or false? Equation Transcription: Text Transcription: Ker(A^2 ) = Ker(A)

True/False Questions Answer Problems 55-60 true or false, and give a brief explanation or counterexample. If A is a square matrix, then \(\operatorname{Im}\left(A^{2}\right)=\operatorname{Im}(A)\). True or false? Equation Transcription: Text Transcription: Im(A^2 ) = Im(A)

True/False Questions Answer Problems 55-60 true or false, and give a brief explanation or counterexample. If A is a square matrix, then \(\operatorname{Ker}(A)=\operatorname{Ker}(R R E F)\). True or false? Equation Transcription: Text Transcription: Ker(A)=Ker(RREF)

True/False Questions Answer Problems 55-60 true or false, and give a brief explanation or counterexample. If A is a square matrix, then \(\operatorname{Im}(A)=\operatorname{Im}(R R E F)\). True or false? Equation Transcription: Text Transcription: Im(A) = Im(RREF)

True/False Questions Answer Problems 55-60 true or false, and give a brief explanation or counterexample. If A and B are \(n \times n\) matrices, then is it true or false that \(\operatorname{Ker}(A+B)=\operatorname{Ker}(A)+\operatorname{Ker}(B)\)? Equation Transcription: Text Transcription: n times n Ker(A+B) = Ker(A) + Ker(B)?

True/False Questions Answer Problems 55-60 true or false, and give a brief explanation or counterexample. Im(A) for \(A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\) is a line in R2. True or false? Equation Transcription: Text Transcription: A=[1 1 _ 1 1]

Detective Work A transformation T: R4 ? R2 is defined with matrix multiplication to be \(T(\bar{v})=A \bar{v}\). It is known that the RREF of A is \(\left[\begin{array}{cccc} 1 & -2 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\) Determine dim(Ker(T)) and dim(Im(T)). Is T one-to-one? Is it onto R2? Find bases for the kernel and image. Equation Transcription: Text Transcription: T(bar v) = A bar v [1 -2 3 0 _ 0 0 0 1]

Detecting Dimensions Consider the transformation T: R2 ? R4 defined by \(T(\bar{v})=B \bar{v}\). The RREF of B is \(\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{array}\right]\) Determine dim(Ker(T)) and dim(Im(T)). Is T one-to-one? Is it onto R4? Equation Transcription: Text Transcription: T(bar v) = B bar v [1 0 _ 0 1 _ 0 0 _ 0 0]

Still Investigating For the transformation T: R3 ? R4 defined by \(T(\bar{v})=A \bar{v}\), where A has RREF \(\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\) determine dim(Ker(T)) and dim(Im(T)). Is T one-to-one? Is it onto R4? Equation Transcription: Text Transcription: T(bar v) = A bar v [1 0 0 _ 0 1 0 _ 0 0 1 _ 0 0 0]

Dimension Theorem Again Consider transformation T: R3 ? R3 defined by \(T(\bar{v})=C \bar{v}\), where C has RREF \(\left[\begin{array}{rrr} 1 & -2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\) Determine dim(Ker(T)) and dim(Im(T)) of transformation T, and decide whether it is injective and/or subjective. Equation Transcription: Text Transcription: T(bar v) = C bar v [1 -2 3 _ 0 0 0 _ 0 0 0]

The Inverse Transformation If T: V ? W is an injective linear transformation, then we can define an inverse transformation T-1: Im(T) ? V so that, for each \(\bar{w}\) in Im(T), T-1\((\bar{w})=\bar{v}\) if and only if \(T(\bar{v})=\bar{w}\). Show that T-1 is an injective and surjective linear transformation. Equation Transcription: Text Transcription: bar w (bar w) = bar v T(bar v) = bar w

Review of Nonhomogeneous Algebraic Systems Express the general solution for each system in Problems 66-71 as the sum of a particular solution and the solution of the corresponding homogeneous system. \(x+y+=1\) Equation Transcription: Text Transcription: x+y=1

Review of Nonhomogeneous Algebraic Systems Express the general solution for each system in Problems 66-71 as the sum of a particular solution and the solution of the corresponding homogeneous system. \(3 x-y+z=-4\) Equation Transcription: Text Transcription: 3x-y+z=-4

Review of Nonhomogeneous Algebraic Systems Express the general solution for each system in Problems 66-71 as the sum of a particular solution and the solution of the corresponding homogeneous system. \(x+2 y=2\) \(2 x+y=2\) Equation Transcription: Text Transcription: x+2y=2 2x+y=2

Review of Nonhomogeneous Algebraic Systems Express the general solution for each system in Problems 66-71 as the sum of a particular solution and the solution of the corresponding homogeneous system. \(x-2 y=5\) \(2 x+4 y=-5\) Equation Transcription: Text Transcription: x-2y=5 2x+4y=-5

Review of Nonhomogeneous Algebraic Systems Express the general solution for each system in Problems 66-71 as the sum of a particular solution and the solution of the corresponding homogeneous system. \(x+2 y-z=6\) \(2 x-y+3 z=-3\) Equation Transcription: Text Transcription: x+2y-z=6 2x-y+3z=-3

Review of Nonhomogeneous Algebraic Systems Express the general solution for each system in Problems 66-71 as the sum of a particular solution and the solution of the corresponding homogeneous system. \(x_{1}+3 x_{2}-4 x_{3}=9\) \(-2 x_{1}+x_{2}+2 x_{3}=-9\) \(-9 x_{1}+15 x_{2}=-3\) Equation Transcription: Text Transcription: x_1 + 3x_2 - 4x_3 = 9 -2x_1 + x_2 + 2x_3 =-9 -9x_1 + 15x_2 =-3

Review of Nonhomogeneous First-Order DEs In each of Problems 72-77, express the general solution of the nonhomogeneous DE as the sum of a particular solution and the general solution of the corresponding homogeneous equation. The homogeneous equations are linear or separable; particular solutions (mostly constant) may be found by inspection. \(y^{\prime}-y=3\) Equation Transcription: Text Transcription: y’ - y =3

Review of Nonhomogeneous First-Order DEs In each of Problems 72-77, express the general solution of the nonhomogeneous DE as the sum of a particular solution and the general solution of the corresponding homogeneous equation. The homogeneous equations are linear or separable; particular solutions (mostly constant) may be found by inspection. \(y^{\prime}+2 y=-1\) Equation Transcription: Text Transcription: y’ + 2y = -1

Review of Nonhomogeneous First-Order DEs In each of Problems 72-77, express the general solution of the nonhomogeneous DE as the sum of a particular solution and the general solution of the corresponding homogeneous equation. The homogeneous equations are linear or separable; particular solutions (mostly constant) may be found by inspection. \(y^{\prime}+\frac{1}{t} y=\frac{1}{t}\) Equation Transcription: Text Transcription: y’ + 1/t y = 1/t

Review of Nonhomogeneous First-Order DEs In each of Problems 72-77, express the general solution of the nonhomogeneous DE as the sum of a particular solution and the general solution of the corresponding homogeneous equation. The homogeneous equations are linear or separable; particular solutions (mostly constant) may be found by inspection. \(y^{\prime}+\frac{1}{t} y=\frac{2}{t^{2}}\) Equation Transcription: Text Transcription: y’ + 1/t^2 y = 2/t^2

Review of Nonhomogeneous First-Order DEs In each of Problems 72-77, express the general solution of the nonhomogeneous DE as the sum of a particular solution and the general solution of the corresponding homogeneous equation. The homogeneous equations are linear or separable; particular solutions (mostly constant) may be found by inspection. \(y^{\prime}+t^{2} y=3 t^{2}\) Equation Transcription: Text Transcription: y’ + t^2 y = 3t^2

Review of Nonhomogeneous First-Order DEs In each of Problems 72-77, express the general solution of the nonhomogeneous DE as the sum of a particular solution and the general solution of the corresponding homogeneous equation. The homogeneous equations are linear or separable; particular solutions (mostly constant) may be found by inspection. \(y^{\prime}+t y=1+t^{2}\) Equation Transcription: Text Transcription: y’ + t y = 1+t^2

Review of Nonhomogeneous Second-Order DEs For each equation in Problems 78-81, express the general solution of the nonhomogeneous DE as the sum of a particular solution (each is a polynomial in t) and the general solution of the corresponding homogeneous DE. \(y^{\prime \prime}+y^{\prime}-2 y=2 t-3\) Equation Transcription: Text Transcription: y’’ + y’ - 2y = 2t -3

Review of Nonhomogeneous Second-Order DEs For each equation in Problems 78-81, express the general solution of the nonhomogeneous DE as the sum of a particular solution (each is a polynomial in t) and the general solution of the corresponding homogeneous DE. \(y^{\prime \prime}-2 y^{\prime}+2 y=4 t-6\) Equation Transcription: Text Transcription: y’’ - 2y’ + 2y = 4t - 6

Review of Nonhomogeneous Second-Order DEs For each equation in Problems 78-81, express the general solution of the nonhomogeneous DE as the sum of a particular solution (each is a polynomial in t) and the general solution of the corresponding homogeneous DE. \(y^{\prime \prime}-2 y^{\prime}+y=t-3\) Equation Transcription: Text Transcription: y’’ - 2y’ + y = t - 3

Review of Nonhomogeneous Second-Order DEs For each equation in Problems 78-81, express the general solution of the nonhomogeneous DE as the sum of a particular solution (each is a polynomial in t) and the general solution of the corresponding homogeneous DE. \(y^{\prime \prime}+y=2 t\) Equation Transcription: Text Transcription: y’’ + y = 2t

Suggested Journal Entry \(I\) The matrix of a linear transformation has been transformed to its reduced row echelon form. Discuss what information about the transformation you can obtain by knowing how many pivots there are and in which rows and columns they appear. Equation Transcription: Text Transcription: I

Suggested Journal Entry II The rows of an \(m \times n\) matrix A, considered as n-vectors, span a subspace of Rn called the row space of A. Its columns span a subspace of Rm called the column space of A. If a linear transformation T: Rn ? Rm is defined by \(T(\bar{v})=A \bar{v}\), discuss the relationship to T of the row and column spaces of A. Equation Transcription: Text Transcription: m times n T(bar v) = A bar v