?(Calculus required) Show that the following sets of functions are subspaces of

In Exercises 1–2, use the Subspace Test to determine which of the sets are subspaces of \(R^{3}\). a. All vectors of the form \((a, 0, 0)\). b. All vectors of the form \((a, 1, 1)\). c. All vectors of the form \((a, b, c)\), where \(b = a + c\). Equation Transcription: Text Transcription: R^3 (a, 0, 0) (a, 1, 1) (a, b, c) b = a + c

In Exercises 1–2, use the Subspace Test to determine which of the sets are subspaces of \(R^{3}\). a. All vectors of the form \((a, b, c)\), where \(b = a + c + 1\). b. All vectors of the form \((a, b, 0)\). c. All vectors of the form \((a, b, c)\) for which \(a + b = 7\). Equation Transcription: Text Transcription: R^3 (a, b, c) b = a + c + 1 (a, b,0) a + b = 7

In Exercises 3–4, use the Subspace Test to determine which of the sets are subspaces of \(M_{n n}\). a. The set of all diagonal \(n \times n\) matrices. b. The set of all \(n \times n\) matrices \(????\) such that \(det(????) = 0\). c. The set of all \(n \times n\) matrices \(????\) such that \(tr(????) = 0\). d. The set of all symmetric \(n \times n\) matrices. Equation Transcription: Text Transcription: M_nn n times n A det(????) = 0 tr(????) = 0

In Exercises 3–4, use the Subspace Test to determine which of the sets are subspaces of \(M_{n n}\). a. The set of all \(n \times n\) matrices \(????\) such that \(A^{t}=A\). b. The set of all \(n \times n\) matrices \(????\) for which \(????x = 0\) has only the trivial solution. c. The set of all \(n \times n\) matrices \(????\) such that \(???????? = ????????\) for some fixed n × n matrix \(B\). d. The set of all invertible \(n \times n\) matrices. Equation Transcription: Text Transcription: M_nn n times n A A^t=-A Ax=0 ???????? = ???????? B

In Exercises 5–6, use the Subspace Test to determine which of the sets are subspaces of \(P_{3}\). a. All polynomials \(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\) for which \(a_{0}=0\). b. All polynomials \(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\) for which \(a_{0}+a_{4}+a_{3}+a_{3}=0\). Equation Transcription: Text Transcription: P_3 a_0+a_1 x+a_2 x^2+a_3 x^3 a_0=0 a_0+a_1+a_2+a_3=0

In Exercises 5–6, use the Subspace Test to determine which of the sets are subspaces of \(P_{3}\). a. All polynomials of the form \(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\) in which \(a_{0}, a_{1}, a_{2}\), and \(a_{0}\) are rational numbers. b. All polynomials of the form \(a_{0}+a_{1} x\), where \(a_{0}\) and \(a_{1}\) are real numbers. Equation Transcription: Text Transcription: P_3 a_0+a_1 x+a_2 x^2+a_3 x^3 a_0,a_1,a_2 a_3 a_0+a_1 x

In Exercises 7–8, use the Subspace Test to determine which of the sets are subspaces of \(F(-\infty, \infty)\). a. All functions \(????\) in \(F(-\infty, \infty)\) for which \(????(0) = 0\). b. All functions \(????\) in \(F(-\infty, \infty)\) for which \(????(0) = 1\). Equation Transcription: Text Transcription: F(-infinity, infinity) f ????(0) = 0 ????(0) = 1

In Exercises 7–8, use the Subspace Test to determine which of the sets are subspaces of \(F(-\infty, \infty)\). a. All functions \(????\) in \(F(-\infty, \infty)\) for which \(????(?x) = ????(x)\). b. All polynomials of degree 2. Equation Transcription: Text Transcription: F(-infinity, infinity) F ????(?x) = ????(x)

In Exercises 9–10, use the Subspace Test to determine which of the sets are subspaces of \(R \infty\). a. All sequences \(v\) in \(R \infty\) of the form \(v = (v, 0, v, 0, v, 0, . . . )\). b. All sequences \(v\) in \(R \infty\) of the form \(v = (v, 1, v, 1, v, 1, . . . )\). Equation Transcription: Text Transcription: R infinity v v = (v, 0, v, 0, v, 0, . . . ) v = (v, 1, v, 1, v, 1, . . . )

In Exercises 9–10, use the Subspace Test to determine which of the sets are subspaces of \(R \infty\). a. All sequences \(v\) in \(R \infty\) of the form \(v = (v, 2v, 4v, 8v, 16v, . . . )\) b. All sequences in \(R \infty\) whose components are 0 from some point on. Equation Transcription: Text Transcription: R infinity v v = (v, 2v, 4v, 8v, 16v, . . . )

In Exercises 11–12, use the Subspace Test to determine which of the sets are subspaces of \(M_{22}\). a. All matrices of the form \(\left[\begin{array}{ll} a & 0 \\ b & 0 \end{array}\right]\) b. All matrices of the form \(\left[\begin{array}{ll} a & 1 \\ b & 1 \end{array}\right]\) c. All 2 × 2 matrices ???? such that \(A\left[\begin{array}{r} 1 \\ -1 \end{array}\right]=\left[\begin{array}{l} 2 \\ 0\end{array}\right]\) Equation Transcription: Text Transcription: M_22 [b 0 \\ a 0] [b 1\\ a 1] A[-1 \\ 1]=[0 \\ 2]

In Exercises 11–12, use the Subspace Test to determine which of the sets are subspaces of \(M_{22}\). a. All \(2 \times 2\) matrices \(????\) such that \(A\left[\begin{array}{r} 1 \\ -1 \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \end{array}\right]\) b. All \(2 \times 2\) matrices \(????\) such that \(A\left[\begin{array}{rr} 0 & 2 \\ -2 & 1 \end{array}\right]=\left[\begin{array}{rr} 0 & 2 \\ -2 & 1 \end{array}\right] A\) c. All \(2 \times 2\) matrices \(????\) for which \(det(????) = 0\). Equation Transcription: Text Transcription: M_22 A[-1 \\ 1]=[0 \\ 0] A[-2 & 1 \\ 0 & 2]=-2 & 1 \\ 0 & 2]A 2 times 2 A det(????) = 0

In Exercises 13–14, use the Subspace Test to determine which of the sets are subspaces of \(R^{4}\). a. All vectors of the form \(\left(a, a^{2}, a^{3}, a^{4}\right)\). b. All vectors of the form \((a, 0, b, 0)\). Equation Transcription: Text Transcription: R^4 (a, a^2, a^3, a^4) (a, 0, b, 0)

In Exercises 13–14, use the Subspace Test to determine which of the sets are subspaces of \(R^{4}\). a. All vectors \(x\) in \(????\) 4 such that \(A x=\left[\begin{array}{l} 0 \\ 1 \end{array}\right]\) where \(A=\left[\begin{array}{rrrr} 0 & -1 & 0 & 2 \\ -1 & 1 & 0 & 1 \end{array}\right]\) b. All vectors x in ???? 4 such that \(A x=\left[\begin{array}{l} 0 \\ 1 \end{array}\right]\) where ???? is as in part (a). Equation Transcription: Text Transcription: x R Ax=[10] A=[-1 & 1 \\ 0 & 1 \\ 0 & -1 \\ 0 & 2] Ax=[1 \\ 0]

In Exercises 15–16, use the Subspace Test to determine which of the sets are subspaces of \(P_{\infty}\) . a. All polynomials of degree less than or equal to 6. b. All polynomials of degree equal to 6. c. All polynomials of degree greater than or equal to 6. Equation Transcription: Text Transcription: P infinity

In Exercises 15–16, use the Subspace Test to determine which of the sets are subspaces of \(P_{\infty}\). a. All polynomials with even coefficients. b. All polynomials whose coefficients sum to 0. c. All polynomials of even degree. Equation Transcription: Text Transcription: P infinity

(Calculus Required) Which of the following are subspaces of \(R_{\infty}\)? a. All sequences of the form \(v=\left(v_{1}, v_{2}, \ldots, v_{n}, \ldots\right)\) such that \(\lim _{n \rightarrow \infty} v_{n}=0\). b. All convergent sequences (that is, all sequences of the form \(v=\left(v_{1}, v_{2}, \ldots, v_{n}, \ldots\right)\) such that \(\lim _{n \rightarrow \infty} v_{n}\) exists). c. All sequences of the form \(v=\left(v_{1}, v_{2}, \ldots, v_{n}, \ldots\right)\) such that \(\sum_{n=1}^{\infty} v_{n}=0\). d. All sequences of the form \(v=\left(v_{1}, v_{2}, \ldots, v_{n}, \ldots\right)\) such that \(\sum_{n=1}^{\infty} v_{n}\) converges. Equation Transcription: Text Transcription: R infinity v = (v_1,v_2 , . . . , v_n, . . .) lim _n right arrow infinity v_n=0 lim _n right arrow infinity v_n sum_n=1^infinity v_n=0 sum_n=1^infinity v_n

A line ???? through the origin in \(R^{3}\) can be represented by parametric equations of the form \(x = at, y = bt\), and \(z = ct\). Use these equations to show that \(L\) is a subspace of \(R_{3}\) by showing that if \(v_{1}=\left(x_{1}, y_{1}, z_{1}\right)\) and \(v_{2}=\left(x_{2}, y_{2^{\prime}} z_{2}\right)\) are points on \(L\) and \(k\) is any real number, then \(kv_{1}\) and \(v_{1}+v_{2}\) are also points on \(L\). Equation Transcription: Text Transcription: x = at, y = bt z = ct R^3 R_3 L k v_1=(x_1,y_1,z_1) v_2=(x_2,y_2,z_2) v_1+v_2

Determine whether the solution space of the system \(????x = 0\) is a line through the origin, a plane through the origin, or the origin only. If it is a plane, find an equation for it. If it is a line, find parametric equations for it. a. \(A=\left[\begin{array}{rrr} -1 & 1 & 1 \\ 3 & -1 & 0 \\ 2 & -4 & -5 \end{array}\right]\) b. \(A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{array}\right]\) c. \(A=\left[\begin{array}{lll} 1 & -3 & 1 \\ 2 & -6 & 2 \\ 3 & -9 & 3 \end{array}\right]\) d. \(A=\left[\begin{array}{rrr} 1 & -1 & 1 \\ 2 & -1 & 4 \\ 3 & 1 & 11 \end{array}\right]\) Equation Transcription: = [] = [] = [] = [] Text Transcription: ????x = 0 A = [3 & -4 & -5\\2 & -1 & 0\\-1 & 1 & 1 ] A = [1 & 0 & 8\\2 & 5 & 3\\1 & 2 & 3 ] A = [3 & -9 & 3\\2 & -6 & 2\\1 & -3 & 1 ] A = [3 & 1 & 11\\2 & -1 & 4\\1 & -1 & 1 ]

(Calculus required) Show that the following sets of functions are subspaces of \(F(-\infty, \infty)\). a. All continuous functions on \((-\infty, \infty)\). b. All differentiable functions on \((-\infty, \infty)\). c. All differentiable functions on \((-\infty, \infty)\) that satisfy \(f^{\prime}+2 f=0\) Equation Transcription: Text Transcription: F(-infinity, infinity) (-infinity, infinity) f'+2f = 0

(Calculus required) Show that the set of continuous functions \(f = f(x)\) on \([a, b]\) such \(t\) \(\int_{a}^{b} f(x) d x=0\) is a subspace of \(c[a, b]\). Equation Transcription: Text Transcription: f = f(x) [a, b] t integral_a^b f(x) dx=0 c[a, b]

Show that the solution vectors of a consistent nonhomogeneous system of m linear equations in \(n\) unknowns do not form a subspace of R^{n} . Equation Transcription: Text Transcription: n R^n

If \(T_{A}\) is multiplication by a matrix \(A\) with three columns, then the kernel of \(T_{A}\) is one of four possible geometric objects. What are they? Explain how you reached your conclusion. Equation Transcription: Text Transcription: T_A A

Consider the following subsets of \(P_{3}: V\) consists of all polynomials \(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\) such that \(a_{0}+a_{3}=0\) and \(W\) consists of all polynomials \(p\) such that \(p(1) = 0\). a. Use the Subspace Test to show that \(????\) and \(????\) are subspaces of \(P_{3}\). b. Show that the set of all polynomials \(p=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\) such that \(a_{0}+a_{3}=0\) and \(p(1) = 0\) is a subspace of \(P_{3}\) without using the Subspace Test. Equation Transcription: Text Transcription: P_3:V a_0+a_1 x+a_2 x^2+a_3 x^3 a_0+a^3=0 V W p p(1)=0 p=a_0+a_1 x+a_2 x^2+a_3 x^3

The accompanying figure shows a mass-spring system in which a block of mass \(m\) is set into vibratory motion by pulling the block beyond its natural position at \(x = 0\) and releasing it at time \(t = 0\). If friction and air resistance are ignored, then the \(x\)-coordinate \(x(t)\) of the block at time t is given by a function of the form \(x(t)=c_{1} \cos \omega t+c_{2} \sin \omega t\) where \(\omega\) is a fixed constant that depends on the mass of the block and the stiffness of the spring and \(c_{1}\) and \(c_{2}\) are arbitrary. Show that this set of functions forms a subspace of \(C \infty(-\infty, \infty)\). Equation Transcription: ?????(??, ?) Text Transcription: m x = 0 t = 0 x x(t) x(t)=c_1 cos omega t+c_2 sin omega t omega c_1 c_2 ???? infinity(-infinity, infinity)

Show that Theorem 4.2.2 would be false if the word “intersection” was replaced with “union” by giving an example of a vector space \(V\) and subspaces \(U\) and \(W\) such that the union of \(U\) with \(W\) is not a subspace of \(V\). Equation Transcription: Text Transcription: V U W

A function \(f = f(x)\) in \(F(-\infty, \infty)\) is even if \(f(?a) = f(a)\) for all real numbers \(a\). Prove that the set of even functions is a subspace of \(F(-\infty, \infty)\). Equation Transcription: Text Transcription: f = f(x) F(-infinity, infinity) ????(?a) = ????(a)

A function \(f = f(x)\) in \(F(-\infty, \infty)\) is odd if \(f(?a) = f(a)\) for all real numbers a. Prove that the set of odd functions is a subspace of \(F(-\infty, \infty)\). Equation Transcription: Text Transcription: f = f(x) F(-infinity, infinity) ????(?a) = ????(a)

If ???? and ???? are subspaces of a vector space \(V\), then the sum of \(U\) and ???? is the set \(U+W\) consisting of all vectors of the form \(u+w\), where u is a vector in \(U\) and w is a vector in \(W\). Prove that \(U+W\) is a subspace of \(V\). Equation Transcription: Text Transcription: V U W ???? + ???? u + w

In parts \((a)–(h)\) determine whether the statement is true or false, and justify your answer. a. Every subspace of a vector space is itself a vector space. Equation Transcription: Text Transcription: (a)–(h)

In parts \((a)–(h)\) determine whether the statement is true or false, and justify your answer. b. Every vector space is a subspace of itself. Equation Transcription: Text Transcription: (a)–(h)

In parts \((a)–(h)\) determine whether the statement is true or false, and justify your answer. Every subset of a vector space \(V\) that contains the zero vector in \(V\) is a subspace of \(V\). Equation Transcription: Text Transcription: (a)–(h) V

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. The kernel of a matrix transformation \(T A: R^{n} \rightarrow R^{m}\) is a subspace of \(R^{m}\). Equation Transcription: Text Transcription: (a)–(h) TA:R^n -> R^m R^m

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. The solution set of a consistent linear system \(Ax = b\) of m equations in \(n\) unknowns is a subspace of \(R^{n}\). Equation Transcription: Text Transcription: Ax=b n R^n

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. The intersection of any two subspaces of a vector space \(V\) is a subspace of \(V\). Equation Transcription: Text Transcription: (a)–(h) V

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. The union of any two subspaces of a vector space \(V\) is a subspace of \(V\). Equation Transcription: Text Transcription: (a)–(h) V

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. The set of upper triangular \(n \times n\) matrices is a subspace of the vector space of all \(n \times n\) matrices. Equation Transcription: Text Transcription: n times n

Determine whether the vectors \(u, v\), and \(w\) are in the kernel of \(T_{A}\), where \(A=\left[\begin{array}{rrrrr} 1 & 2 & 3 & 4 & 5 \\ 6 & 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 & 15 \\ 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 & 25 \end{array}\right]\) and \(u = (1, ?2, 1, 0, 0), v = (5, 0, 1, ?2, 1), w = (3, ?4, 0, 0, 1)\) Equation Transcription: A=[ ] Text Transcription: u v w T_A u = (1, ?2, 1, 0, 0), v = (5, 0, 1, ?2, 1), w = (3, ?4, 0, 0, 1) A= [1 & 2 & 3 & 4 & 5 \\ 6 & 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 & 15 \\ 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 & 25 ]