?In Exercises 3–9, determine whether the mapping ???? is a linear transformation, and if

In Exercises 1–2, suppose that ???? is a mapping whose domain is the vector space????22. In each part, determine whether ???? is a linear transformation, and if so, find its kernel. a. \(T(A)=A^{2}\) b. \(T(A)=\operatorname{tr}(A)\) c. \(T(A)=A+A^{T}\) Equation Transcription: ????(????) = ????2 ????(????) = tr(????) ????(????) = ???? + ???????? Text Transcription: T(A) = A^2 T(A) = tr(A) T(A) = A + A^T

In Exercises 1–2, suppose that ???? is a mapping whose domain is the vector space????22. In each part, determine whether ???? is a linear transformation, and if so, find its kernel. a. \(T(A)=(A)^{11}\) b. \(T(A)=0_{2 \times 2}\) c. \(T(A)=\mathrm{C} A\) Equation Transcription: ????(????) = (????)11 ????(????) = 02×2 ????(????) = c???? Text Transcription: T(A) = (A)^11 T(A) = 0_2x2 T(A) = cA

In Exercises 3–9, determine whether the mapping ???? is a linear transformation, and if so, find its kernel. \(T: R^{3} \rightarrow R, \text { where } T(\mathrm{u})=\|\mathrm{u}\|\). Equation Transcription: ???? ? ????3 ????, where ????(u) = ||u|| Text Transcription: T ? R^3 rightarrow R, where ????(u) = ||u||

In Exercises 3–9, determine whether the mapping ???? is a linear transformation, and if so, find its kernel. \(T: R^{3} \rightarrow R^{3} \text {, where } v_{0} \text { is a fixed vector in } R^{3} \text { and } T(u)=u \times v_{0}\) Equation Transcription: ???? ? ????3 ????3 , where v0 is a fixed vector in ???? 3 and ????(u) = u v0 Text Transcription: T ? R^3 rightarrow R^3 , where v_0 is a fixed vector in R^3 and T(u) = u v_0

In Exercises 3–9, determine whether the mapping ???? is a linear transformation, and if so, find its kernel. \(T: M_{22} \rightarrow M_{23}, \text { where } B \text { is a fixed } 2 \times 3 \text { matrix and } T(A)=A B\) Equation Transcription: ???? ? ????22 ????23, where ???? is a fixed 2 3 matrix and ????(????) = ???????? Text Transcription: T ? M_22 M_23, where B is a fixed 2 x 3 matrix and T(A) = AB

In Exercises 3–9, determine whether the mapping ???? is a linear transformation, and if so, find its kernel. \(T: M_{22} \rightarrow R\), where a. \(T\left(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\right)=3 a-4 b+c-d \) b. \(T\left(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\right)=a^{2}+b^{2} \) Equation Transcription: ???? ? ????22 ???? [] [] Text Transcription: T ? M_22 rightarrow R T([a b c d])=3a-4b+c-d T([a b c d])=a^2+b^2

In Exercises 3–9, determine whether the mapping ???? is a linear transformation, and if so, find its kernel. \(T: P_{2} \rightarrow P_{2}\) , where a. \(T\left(a_{0}+a_{1} x+a_{2} x^{2}\right)=a_{0}+a_{1}(x+1)+a_{2}(x+1)^{2}\) b. \(T\left(a_{0}+a_{1} x+a_{2} x^{2}\right)=\left(a_{0}+1\right)+\left(a_{1}+1\right) x+\left(a_{2}+1\right) x^{2}\) Equation Transcription: ???? ? ????2 ????2 ????(a0 + a1 x + a2 x 2 ) = a0 + a1 (x + 1) + a2 (x + 1) 2 ????(a0 + a1 x + a2 x 2 ) = (a0 + 1) + (a1 + 1)x + (a2 + 1)x 2 Text Transcription: T ? P2 rightarrow P2 T(a_0 + a_1 x + a_2 x^2 ) = a_0 + a_1 (x + 1) + a_2 (x + 1)^2 T(a_0 + a_1 x + a_2 x^2 ) = (a_0 + 1) + (a_1 + 1)x + (a_2 + 1)x^2

In Exercises 3–9, determine whether the mapping ???? is a linear transformation, and if so, find its kernel. \(T: F(-\infty, \infty) \rightarrow F(-\infty, \infty)\), where a. \(T(f(\mathrm{x}))=1+f(\mathrm{x})\) b. \(T(f(x))=f(x+1)\) Equation Transcription: ???? ? ????(, ) ????(, ) ????(????(x)) = 1 + ????(x) ????(????(x)) = ????(x + 1) Text Transcription: T ? F(-infinity, infinity) rightarrow F(-infinity, infinity) ????(????(x)) = 1 + ????(x) ????(????(x)) = ????(x + 1)

In Exercises 3–9, determine whether the mapping ???? is a linear transformation, and if so, find its kernel. \(\mathrm{T}: \mathrm{R}^{\infty} \rightarrow \mathrm{R}^{\infty}\), where \(T\left(\mathrm{a}_{0}, \mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{\mathrm{n}}, \ldots\right)=\left(0, \mathrm{a}_{0}, \mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{\mathrm{n}}, \ldots\right)\) Equation Transcription: T ? R R ????(a0 , a1 , a2 , . . . , an , . . . ) = (0, a0 , a1 , a2 , . . . , an , . . . ) Text Transcription: T ? R infinity rightarrrow R infinity T(a_0 , a_1 , a_2 , . . . , a_n , . . . ) = (0, a_0 , a_1 , a_2 , . . . , a_n , . . . )

Let \(T: P_{2} \rightarrow P_{3}\) be the linear transformation defined by \(T(\mathrm{p}(\mathrm{x}))=\mathrm{xp}(\mathrm{x})\). Which of the following are in ker(????)? a. \(x^{2}\) b. \(0\) c. \(1 + x\) d. \(?x\) Equation Transcription: ???? ? ????2 ????3 ????(p(x)) = xp(x) x 2 0 1 + x ?x Text Transcription: T ? P2 rightarrow P3 ????(p(x)) = xp(x) x^2 0 1 + x ?x

Let \(T: P_{2} \rightarrow P_{3}\) be the linear transformation in Exercise 10. Which of the following are in ????(????)? a. \(x+x^{2}\) b. \(1+x\) c. \(3-x^{2}\) d. \(-x\) Equation Transcription: ???? ? ????2 ????3 x + x 2 1 + x 3 ? x 2 ?x Text Transcription: T ? P2 rightarrow P3 x + x^2 1 + x 3 ? x^2 ?x

Let ???? be any vector space, and let \(T: V \rightarrow V\) be defined by \(\mathrm{T}(\mathrm{V})=3 \mathrm{v}\). a. What is the kernel of ????? b. What is the range of ????? Equation Transcription: T ? V V T(V) = 3v Text Transcription: T ? V rightarrow V T(V) = 3v

In each part, use the given information to find the nullity of the linear transformation ????. a. \(T: R^{5} \rightarrow P_{5}\) has rank 3. b. \(T: P_{4} \rightarrow P_{3}\) has rank 1. c. The range of \(T: M_{\mathrm{mn}} \rightarrow R^{3} \text { is } R^{3}\). d. \(T: M_{22} \rightarrow M_{22}\) has rank 3. Equation Transcription: ???? ?????5 ????5 ???? ? ????4 ????3 ???? ? ????mn ????3 is ???? 3 ???? ? ????22 ????22 Text Transcription: T ?R^5 rightarrow P_5 T ? P_4 rightarrow P_3 T ? M_mn rightarrow R^3 is R^3 T ? M_22 rightarrow M_22

In each part, use the given information to find the rank of the linear transformation ????. a. \(T: R^{7} \rightarrow M_{32}\) has nullity 2. b. \(T: P_{3} \rightarrow R\) has nullity 1. c. The null space of \(T: P_{5} \rightarrow P_{5} \text { is } P_{5}\) . d. \(T: P_{\mathrm{n}} \rightarrow M_{\mathrm{mn}}\) has nullity 3. Equation Transcription: ???? ?????7 ????32 ???? ? ????3 ???? ???? ? ????5 ????5 is ????5 ???? ? ????n ????mn Text Transcription: T ?R^7 rightarrow M_32 T ? P_3 rightarrow R T ? P_5 rightarrow P_5 is P_5 T ? P_n rightarrow M_mn

Let \(T: M_{22} \rightarrow M_{22}\) be the dilation operator with factor k = 3. a. Find \(T\left(\left[\begin{array}{ll} 1 & 2 \\ -4 & 3 \end{array}\right]\right) \) b. Find the rank and nullity of ????. Equation Transcription: ???? ? ????22 ????22 [] Text Transcription: T:M_22 rightarrow M_22 T([1 2 -4 3])

Let \(T: P_{2} \rightarrow P_{2}\) be the contraction operator with factor \(k=1 / 4\). a. Find \(T\left(1+4 x+8 x^{2}\right)\). b. Find the rank and nullity of ????. Equation Transcription: ???? ? ????2 ????2 ????(1 + 4x + 8x 2) Text Transcription: T ? P_2 rightarrow P_2 k = 1/4 T(1 + 4x + 8x^2)

Let \(T: P_{2} \rightarrow P_{3}\) be the evaluation transformation at the sequence of points \(-1,0,1\). Find a. \(T\left(\mathrm{x}^{2}\right)\) b. \(\operatorname{ker}(T)\) c. \(R(T)\) Equation Transcription: ???? ? ????2 ????3 ????(x 2) ker(????) ????(????) Text Transcription: T ? P_2 rightarrow P_3 -1, 0, 1 T(x*2) ker(T) R(T)

Let ???? be the subspace of \(C[0,2 \pi]\) spanned by the vectors 1, sin x, and cos x, and let \(T: V \rightarrow R^{3}\) be the evaluation transformation at the sequence of points 0, \(\pi\), \(2 \pi\). Find a. \(T(1+\sin x+\cos x)\) b. \(\operatorname{ker}(T)\) c. \(R(T)\) Equation Transcription: ???? ? ???? ????3 Text Transcription: C[0, 2 pi] T ?V rightarrow R^3 pi 2 pi T(1 + sin x + cos x) ker(T) R(T)

Consider the basis \(S=\left\{\mathrm{v}_{1}, \mathrm{v}_{2}\right\}\) for \(R^{2}\) , where \(v_{1}=(1,1)\) and \(v_{2}=(1,0)\), and let \(T: R^{2} \rightarrow R^{2}\) be the linear operator for which \(T\left(v_{1}\right)=(1,-2) \text { and } T\left(v_{2}\right)=(-4,1)\) Find a formula for \(T\left(x_{1}, x_{2}\right)\), and use that formula to find \(T(5,-3)\). Equation Transcription: ???? = {v1 , v2 } ????2 v1 = (1, 1) v2 = (1, 0) ???? ?????2 ????2 ????(v1 ) = (1, ?2) and ????(v2 ) = (?4, 1) ????(x1 , x2 ) ????(5, ?3) Text Transcription: S = {v_1 , v_2 } R^2 v_1 = (1, 1) v_2 = (1, 0) T ?R^2 rightarrow R^2 T(v_1 ) = (1, ?2) and T(v_2 ) = (?4, 1) T(x_1 , x_2 ) T(5, ?3)

Consider the basis \(S=\left\{v_{1}, v_{2}\right\}\) for \(R^{2}\) , where \(v_{1}=(-2,1)\) and \(\mathrm{v}_{2}=(1,3)\), and let \(T: R^{2} \rightarrow R^{3}\) be the linear transformation such that \(T\left(\mathrm{v}_{1}\right)=(-1,2,0) \text { and } T\left(\mathrm{v}_{2}\right)=(0,-3,5)\) Find a formula for \(T\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)\), and use that formula to find \(T(2,-3)\). Equation Transcription: ???? = {v1 , v2 } ????2 v1 = (?2, 1) v2 = (1, 3) ???? ?????2 ????3 ????(v1 ) = (?1, 2, 0) and ????(v2 ) = (0, ?3, 5) ????(x1 , x2 ) ????(2, ?3) Text Transcription: S = {v_1 , v_2 } R^2 v_1 = (?2, 1) v_2 = (1, 3) T ?R^2 rightarrow R^3 T(v_1 ) = (?1, 2, 0) and T(v_2 ) = (0, ?3, 5) T(x_1 , x_2 ) T(2, ?3)

Consider the basis \(S=\left\{\mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}\right\}\) for \(R^{3}\) , where \(\mathrm{v}_{1}=(1,1,1), \mathrm{v}_{2}=(1,1,0)\), and \(v_{3}=(1,0,0)\), and let \(T: R^{3} \rightarrow R^{3}\) be the linear operator for which \(T\left(\mathrm{v}_{1}\right)=(2,-1,4), T\left(\mathrm{v}_{2}\right)=(3,0,1), T\left(\mathrm{v}_{3}\right)=(-1,5,1)\) Find a formula for \(T\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}\right)\), and use that formula to find \(T(2,4,-1)\). Equation Transcription: ???? = {v1 , v2 , v3 } ????3 v1 = (1, 1, 1), v2 = (1, 1, 0) v3 = (1, 0, 0) ???? ?????3 ????3 ????(v1 ) = (2, ?1, 4), ????(v2 ) = (3, 0, 1), ????(v3 ) = (?1, 5, 1) ????(x1 , x2 , x3 ) ????(2, 4, ?1) Text Transcription: S = {v_1 , v_2 , v_3 } R^3 v_1 = (1, 1, 1), v_2 = (1, 1, 0) v_3 = (1, 0, 0) T ?R^3 rightarrow R^3 T(v_1 ) = (2, ?1, 4), T(v_2 ) = (3, 0, 1), T(v_3 ) = (?1, 5, 1) T(x_1 , x_2 , x_3 ) T(2, 4, ?1)

Consider the basis \(S=\left\{\mathrm{V}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}\right\}\) for \(R^{3}\) , where \(v_{1}=(1,2,1), v_{2}=(2,9,0)\), and \(\mathrm{v}_{3}=(3,3,4)\), and let \(T: R^{3} \rightarrow R^{2}\) be the linear transformation for which \(T\left(v_{1}\right)=(1,0), T\left(v_{2}\right)=(-1,1), T\left(v_{3}\right)=(0,1)\) Find a formula for \(T\left(x_{1}, x_{2}, x_{3}\right)\), and use that formula to find \(T(7,13,7)\). Equation Transcription: ???? = {v1 , v2 , v3 } ????3 v1 = (1, 2, 1), v2 = (2, 9, 0) v3 = (3, 3, 4) ???? ?????3 ????2 ????(v1 ) = (1, 0), ????(v2 ) = (?1, 1), ????(v3 ) = (0, 1) ????(x1 , x2 , x3 ) ????(7, 13, 7) Text Transcription: S = {v_1 , v_2 , v_3 } R^3 v_1 = (1, 2, 1), v_2 = (2, 9, 0) v_3 = (3, 3, 4) T ?R^3 rightarrow R^2 T(v_1 ) = (1, 0), T(v_2 ) = (?1, 1), T(v_3 ) = (0, 1) T(x_1 , x_2 , x_3 ) T(7, 13, 7)

In Exercises 23–24, let ???? be multiplication by the matrix ????. Find a. a basis for the range of ????. b. a basis for the kernel of ????. c. the rank and nullity of ????. d. the rank and nullity of ????. \(A=\left[\begin{array}{ccc} 1 & -1 & 3 \\ 5 & 6 & -4 \\ 7 & 4 & 2 \end{array}\right] \) Equation Transcription: A=[] Text Transcription: A=[1 -1 3 5 6 -4 7 4 2]

In Exercises 23–24, let ???? be multiplication by the matrix ????. Find a. a basis for the range of ????. b. a basis for the kernel of ????. c. the rank and nullity of ????. d. the rank and nullity of ????. \(A=\left[\begin{array}{ccc} 2 & 0 & -1 \\ 4 & 0 & -2 \\ 20 & 0 & 0 \end{array}\right] \) Equation Transcription: A=[] Text Transcription: A=[2 0 -1 4 0 -2 20 0 0]

In Exercises 25–26, let \(T_{A}: R^{4} \rightarrow R^{3}\) be multiplication by ????. Find a basis for the kernel of \(T_{A}\), and then find a basis for the range of \(T_{A}\) that consists of column vectors of ????. \(A=\left[\begin{array}{cccc} 1 & 2 & -1 & -2 \\ -3 & 1 & 3 & 4 \\ -3 & 8 & 4 & 2 \end{array}\right] \) Equation Transcription: ???????? ?????4 ????3 ???????? A=[] Text Transcription: T_A:R^4 rightarrow R^3 T_A A=[1 2 -1 -2 -3 1 3 4 -3 8 4 2]

In Exercises 25–26, let \(T_{A}: R^{4} \rightarrow R^{3}\) be multiplication by ????. Find a basis for the kernel of \(T_{A}\), and then find a basis for the range of \(T_{A}\) that consists of column vectors of ????. \(A=\left[\begin{array}{cccc} 1 & 1 & 0 & 1 \\ -2 & 4 & 2 & 2 \\ -1 & 8 & 3 & 5 \end{array}\right] \) Equation Transcription: ???????? ?????4 ????3 ???????? A=[] Text Transcription: T_A:R^4 rightarrow R^3 T_A A=[1 1 0 1 -2 4 2 2 -1 8 3 5]

Let \(T: P_{3} \rightarrow P_{2}\) be the mapping defined by \(T\left(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\right)=5 a_{0}+a_{3} x^{2}\) a. Show that ???? is linear. b. Find a basis for the kernel of ????. c. Find a basis for the range of ????. Equation Transcription: ???? ? ????3 ????2 ????(a0 + a1 x + a2 x 2 + a3 x 3 ) = 5a0 + a3 x 2 Text Transcription: T? P3 rightarrow P2 T(a_0 + a_1 x + a_2 x^2 + a_3 x^3 ) = 5a_0 + a_3 x^2

Let \(T: P_{2} \rightarrow P_{2}\) be the mapping defined by \(T\left(a_{0}+a_{1} x+a_{2} x^{2)}=3 a_{0}+a_{1} x+\left(a_{0}+a_{1}\right) x^{2}\right.\) a. Show that ???? is linear. b. Find a basis for the kernel of ????. c. Find a basis for the range of ????. Equation Transcription: ???? ? ????2 ????2 ????(a0 + a1 x + a2 x 2 ) = 3a0 + a1 x + (a0 + a1 )x 2 Text Transcription: T? P2 rightarrow P2 T(a_0 + a_1 x + a_2 x^2 ) = 3a_0 + a_1 x + (a_0 + a_1 )x^2

a. (Calculus required) Let \(D: P_{3} \rightarrow P_{2}\) be the differentiation transformation \(D(p)=p^{\prime}(x)\). What is the kernel of ????? b. (Calculus required) Let \(J: P_{1} \rightarrow R\) be the integration transformation \(J(p)=\int_{-1}^{1} p(x) d x\). What is the kernel of ????? Equation Transcription: ???? ? ????3 ????2 ????(p) = p?(x) ???? ? ????1 ???? ? Text Transcription: D ? P_3 rightarrow P_2 D(p) = p?(x) J ? P_1 rightarrow R J(p)= Integral_-1^1 p(x) dx

(Calculus required) Let \(V=C[a, b]\) be the vector space of continuous functions on \([a, b]\), and let \(\mathrm{T}: V \rightarrow V\) be the transformation defined by \(T(f)=5 f(x)+3 \int_{a}^{x} f(t) d t\) Is ???? a linear operator? Equation Transcription: ???? = C[a, b] [a, b] T ? ???? ???? ? Text Transcription: ???? = C[a, b] [a, b] T ? V rightarrow V T(f)=5f(x)+3 Integral_a^x f(t) dt

(Calculus required) Let ???? be the vector space of real-valued functions with continuous derivatives of all orders on the interval \((-\infty, \infty)\), and let \(W=F(-\infty, \infty)\) be the vector space of real-valued functions defined on (-\infty, \infty). a. Find a linear transformation ???? ? ???? ????? whose kernel is ????3 . b. Find a linear transformation ???? ? ???? ? ???? whose kernel is ????n . Equation Transcription: (, ) ???? = ???? (, ) Text Transcription: (-infinity, infinity) ???? = ???? (-infinity, infinity)

For a positive integer \(n>1\), let \(T: M_{\mathrm{nn}} \rightarrow R\) be the linear transformation defined by \(T(A)=\operatorname{tr}(A)\), where ???? is an \(n \times n\) matrix with real entries. Determine the dimension of ker(????). Equation Transcription: n > 1 ???? ? ????nn ???? ????(????) = tr(????) Text Transcription: n > 1 T ? M_nn rightarrow R T(????) = tr(????) n x n

a. Let \(T: V \rightarrow R^{3}\) be a linear transformation from a vector space ???? to \(R^{3}\) . Geometrically, what are the possibilities for the range of ????? b. Let \(T: R^{3} \rightarrow W\) be a linear transformation from \(R^{3}\) to a vector space ????. Geometrically, what are the possibilities for the kernel of ????? Equation Transcription: ???? ? ???? ????3 ????3 ???? ?????3 ???? Text Transcription: T ? V rightarrow R^3 R^3 T ?R^3 rightarrow ????

In each part, determine whether the mapping \(T: P_{\mathrm{n}} \rightarrow P_{\mathrm{n}}\) is linear. a. \(T(p(x))=p(x+1)\) b. \(T(p(x))=p(x)+1\) Equation Transcription: ???? ? ????n ????n ????(p(x)) = p(x + 1) ????(p(x)) = p(x) + 1 Text Transcription: T ? Pn rightarrow Pn T(p(x)) = p(x + 1) T(p(x)) = p(x) + 1

Let \(\mathrm{v}_{1}, \mathrm{v}_{2}\) , and \(\mathrm{v}_{3}\) be vectors in a vector space ????, and let \(T: V \rightarrow R^{3}\) be a linear transformation for which \(T\left(\mathrm{v}_{1}\right)=(1,-1,2), T\left(\mathrm{v}_{2}\right)=(0,3,2), T\left(\mathrm{v}_{3}\right)=(-3,1,2)\) Find \(T\left(2 v_{1}-3 v_{2}+4 v_{3}\right)\). Equation Transcription: v1 , v2 v3 ???? ? ???? ????3 ????(v1 ) = (1, ?1, 2), ????(v2 ) = (0, 3, 2), ????(v3 ) = (?3, 1, 2) ????(2v1 ? 3v2 + 4v3 ) Text Transcription: v_1 , v_2 v_3 T ? V rightarrow R^3 T(v_1 ) = (1, ?1, 2), T(v_2 ) = (0, 3, 2), T(v_3 ) = (?3, 1, 2) T(2v_1 ? 3v_2 + 4v_3 )

Let \(\left\{\mathrm{v}_{1}, \mathrm{v}_{2}, \ldots, \mathrm{v}_{\mathrm{n}}\right\}\) be a basis for a vector space ????, and let \(T: V \rightarrow V\) be a linear transformation. Prove that if \(T\left(\mathrm{v}_{1}\right)=T\left(\mathrm{v}_{2}\right)=\cdots=T\left(\mathrm{~V}_{\mathrm{n}}\right)=0\) then ???? is the zero transformation. Equation Transcription: {v1 , v2 , . . . , vn } ???? ? ???? ???? ????(v1 ) = ????(v2 ) = ? ? ? = ????(vn ) = 0 Text Transcription: {v_1 , v_2 , . . . , v_n } T ? V rightarrow V T(v_1 ) = T(v_2 ) = ? ? ? = ????(v_n ) = 0

Let \(\left\{\mathrm{v}_{1}, \mathrm{v}_{2}, \ldots, \mathrm{v}_{\mathrm{n}}\right\}\) be a basis for a vector space ????, and let \(T: V \rightarrow V\) be a linear operator. Prove that if \(T\left(v_{1}\right)=v_{1}, T\left(v_{2}\right)=v_{2}, \ldots, T\left(v_{n}\right)=v_{n}\) then ???? is the identity transformation on ????. Equation Transcription: {v1 , v2 , . . . , vn } ???? ? ???? ???? ????(v1 ) = v1 , ????(v2 ) = v2 , . . . , ????(vn ) = vn Text Transcription: {v_1 , v_2 , . . . , v_n } T ? V rightarrow V T(v_1 ) = v_1 , T(v_2 ) = v_2 , . . . , T(v_n ) = v_n

Prove: If \(\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}\) is a basis for a vector space ???? and \(\mathrm{w}_{1}, \mathrm{w}_{2}, \ldots, \mathrm{w}_{\mathrm{n}}\) are vectors in a vector space ????, not necessarily distinct, then there exists a linear transformation ???? that maps ???? into ???? such that \(T\left(\mathrm{v}_{1}\right)=\mathrm{w}_{1}, T\left(\mathrm{v}_{2}\right)=\mathrm{w}_{2}, \ldots, T\left(\mathrm{v}_{\mathrm{n}}\right)=\mathrm{w}_{\mathrm{n}}\) Equation Transcription: {v1 , v2 , . . . , vn } w1 , w2 , . . . , wn ????(v1 ) = w1 , ????(v2 ) = w2 , . . . , ????(vn ) = wn Text Transcription: {v_1 , v_2 , . . . , v_n } w_1 , w_2 , . . . , w_n T(v_1 ) = w_1 , T(v_2 ) = w_2 , . . . , T(v_n ) = w_n

Let \(\mathrm{q}_{0}(\mathrm{x})\) be a fixed polynomial of degree m, and define a function ???? with domain \(P_{\mathrm{n}}\) by the formula \(T(p(x))=p\left(q_{0}(x)\right)\). Prove that ???? is a linear transformation. Equation Transcription: q0 (x) ????n ????(p(x)) = p(q0 (x)) Text Transcription: q_0 (x) P_n T(p(x)) = p(q_0 (x))

In parts \((a)–(i)\) determine whether the statement is true or false, and justify your answer. If \(T\left(\mathrm{c}_{1} \mathrm{v}_{1}+\mathrm{C}_{2} \mathrm{v}_{2}\right)=\mathrm{c}_{1} T\left(\mathrm{v}_{1}\right)+\mathrm{c}_{2} T\left(\mathrm{v}_{2}\right)\) for all vectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) in ???? and all scalars \(\mathrm{c}_{1}\) and \(\mathrm{c}_{2}\) , then ???? is a linear transformation. Equation Transcription: ????(c1v1 + c2v2 ) = c1????(v1 ) + c2????(v2 ) v1 v2 c1 c2 Text Transcription: ????(c1v1 + c2v2 ) = c1????(v1 ) + c2????(v2 ) v1 v2 c1 c2

In parts \((a)–(i)\) determine whether the statement is true or false, and justify your answer. If v is a nonzero vector in ????, then there is exactly one linear transformation \(T: V \rightarrow W\) such that \(\mathrm{T}(-\mathrm{v})=-\mathrm{T}(\mathrm{v})\) Equation Transcription: T ? V W T(?v) = ?T(v) Text Transcription: (a)–(i) T ? V rightarrow W T(?v) = ?T(v)

In parts \((a)–(i)\) determine whether the statement is true or false, and justify your answer. There is exactly one linear transformation \(T: V \rightarrow W\) for which \(T(u+v)=T(u-v)\) for all vectors u and v in ????. Equation Transcription: T ? V W T(u + v) = T(u ? v) Text Transcription: (a)–(i) T ? V rightarrow W T(u + v) = T(u ? v)

In parts (\((a)–(i)\) determine whether the statement is true or false, and justify your answer. If \(\mathrm{v}_{0}\) is a nonzero vector in ????, then \(T(v)=v_{0}+v\) defines a linear operator on ????. Equation Transcription: v0 T(v) = v0 + v Text Transcription: (a)–(i) v_0 T(v) = v_0 + v

In parts \((a)–(i)\) determine whether the statement is true or false, and justify your answer. The kernel of a linear transformation is a vector space. Equation Transcription: Text Transcription: (a)–(i)

In parts \((a)–(i)\) determine whether the statement is true or false, and justify your answer. The range of a linear transformation is a vector space. Equation Transcription: Text Transcription: (a)–(i)

In parts \((a)–(i)\) determine whether the statement is true or false, and justify your answer. If \(\mathrm{T}: P_{6} \rightarrow M_{22}\) is a linear transformation, then the nullity of ???? is 3. Equation Transcription: T ? ????6 ????22 Text Transcription: (a)–(i) T ? P_6 rightarrow M_22

In parts \((a)–(i)\) determine whether the statement is true or false, and justify your answer. The function \(\mathrm{T}: M_{22} \rightarrow R\) defined by \(\mathrm{T}(\mathrm{A})=\operatorname{det} \mathrm{A}\) is a linear transformation. Equation Transcription: T ? ????22 ???? T(A) = det A Text Transcription: (a)–(i) T ? M_22 rightarrow R T(A) = det A

In parts \((a)–(i)\) determine whether the statement is true or false, and justify your answer. The linear transformation \(T: M_{22} \rightarrow M_{22}\)defined by \(T(A)=\left[\begin{array}{ll}1 & 3 \\ 2 & 6 \end{array}\right] \) has rank 1. Equation Transcription: ???? ? ????22 ????22 T(A)=[] Text Transcription: (a)–(i) T ? M_22 rightarrow M_22 T(A)=[1 3 2 6]