Assignment: Vector Spaces and Properties
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(1) Determine whether the following set of vectors are vector spaces or not over \(R\), with respect to given addition ' + ' and scalar multiplication '.'.
(a) The set \[ \left\{(a, b) \in \mathbb{R}^2: b=5 a+1\right\}, \] with respect to usual addition and multiplication.
(b) The set \(M_n(\mathbb{R})\) of all \(n \times n\) real matrices over \(\mathbb{R}\) with respect to usual addition and multiplication.
(c) The set \(\mathbb{R}^2\) over \(\mathbb{R}\) with respect to the addition and scalar multiplication defined by \( \lambda \cdot(a, b):=(\lambda a, 0) \) for \((a, b) \in \mathbb{R}^2, \lambda \in \mathbb{R}\).
(d) The set \(C[a, b]\) of all real valued continuous functions defined on the interval \([a, b]\), with respect to the following addition and scalar multiplication: \((f+g)(t):=f(t)+g(t)\) and \((\lambda \cdot f)(t):=\lambda f(t)\) for \(f, g \in C[a, b], t \in[a, b], \lambda \in \mathbb{R}\).
(e) The set of all positive real numbers over \(\mathbb{R}\), with respect to the following addition and scalar multiplication: \(x+y:=x y\) and \(\lambda \cdot x:=x^\lambda\) for \(x, y \in \mathbb{R}, \lambda \in \mathbb{R}\).
(f) The set \(P_n\) of all real polynomials of order less than or, equal to \(n\), with respect to usual addition and scalar multiplication.
(2) Let \(V=\left\{\left(a_1, a_2\right): a_1, a_2 \in \mathbb{R}\right\}\). For \(\left(a_1, a_2\right),\left(b_1, b_2\right) \in V\) and \(c \in \mathbb{R}\), define
\( \left(a_1, a_2\right)+\left(b_1, b_2\right)=\left(a_1+2 b_1, a_2+3 b_2\right) \text { and } c\left(a_1, a_2\right)=\left(c a_1, c a_2\right) \text {. } \)
Is \(V\) a vector space over \(\mathbb{R}\) with these operations? Justify your answer.
(3) Determine whether the following sets are subspaces of \(R^3\) under the operations of addition and scalar multiplication defined on \mathbb{R}^3\). Justify your answers.
(a) \(W_1=\left\{\left(a_1, a_2, a_3\right) \in \mathbb{R}^3: a_1=3 a_2\right.\) and \(\left.a_3=-a_2\right\}\)
(b) \(W_2=\left\{\left(a_1, a_2, a_3\right) \in \mathbb{R}^3: 2 a_1-7 a_2+a_3=0\right\}\)
(c) \(W_3=\left\{\left(a_1, a_2, a_3\right) \in \mathbb{R}^3: a_1+2 a_2-3 a_3=1\right\}\)
(d) \(W_4=\left\{\left(a_1, a_2, a_3\right) \in \mathbb{R}^3: 5 a_1^2-3 a_2^2+6 a_3^2=0\right\}\)
(4) Describe the smallest subspace of the space \(M_2(\mathbb{R})\) containing \( \left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right)\) and \(\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)\). Justify your answer.
(5) Determine whether or not the set \(\left\{f \in C[0,1]: \int_0^1 f(x) d x=0\right\}\) is a subspace of \(C[0,1]\).
(6) Show that the subspaces of \(\mathbb{R}^3\) are precisely \(\{0\}, \mathbb{R}^3\), all lines in \(\mathbb{R}^3\) through the origin, and all planes in \(\mathbb{R}^3\) through the origin.
(7) If \(U_1, U_2\) are two subspaces of a vector space \(V\) then \(U_1+U_2\) is the direct sum \(U_1 \oplus U_2\) of \(U_1, U_2\) if every element \(u\) of \(U_1+U_2\) can be uniquely written as \(u=u_1+u_2\) where \(u_1 \in U_1\) and \(u_2 \in U_2\).
(a) Prove or give a counterexample: if \(U_1, U_2, W\) are subspaces of a vector space \(V\) such that \(V=U_1 \oplus W=U_2 \oplus W\) then \(U_1=U_2\).
(b) Let \(U_e\) denote the set of real-valued even functions on \(\mathbb{R}\) and let \(U_o\) denote the set of real-valued odd functions on \(\mathbb{R}\). Show that \(\mathbb{R}^{\mathbb{R}}=U_e \oplus U_o\), where \(\mathbb{R}^{\mathbb{R}}\) denotes the space of all functions from \(\mathbf{R}\) to \(\mathbf{R}\).
(8) For each of the following lists of vectors in \(\mathbb{R}^3\), determine whether the first vector can be expressed as a linear combination of the other two.
(a) \((-2,0,3),(1,3,0),(2,4,-1)\)
(b) \((3,4,1),(1,-2,1),(-2,-1,1)\)
(c) \((5,1,-5),(1,-2,-3),(-2,3,-4)\)
(9) Determine whether the vectors \(v_1=(1,-1,4), v_2=(-2,1,3)\) and \(v_3=(4,-3,5)\) span \(\mathbf{R}^3\).
(10) Verify that \(A_1=\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right), A_2=\left(\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right), A_3=\left(\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right)\), and \(A_4=\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right)\) span \(M_2(\mathbb{R})\).
(11) Show that if
\( M_1=\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right), \quad M_2=\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right), \quad \text { and } \quad M_3=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \)
then the span of \(\left\{M_1, M_2, M_3\right\}\) is the set of all \(2 \times 2\) symmetric matrices.
(12) Determine whether the following sets are linearly dependent or linearly independent.
(a) \(\left\{\left(\begin{array}{rr}1 & -3 \\ -2 & 4\end{array}\right),\left(\begin{array}{rr}2 & -1 \\ 3 & 2\end{array}\right),\left(\begin{array}{rr}1 & -8 \\ -9 & 10\end{array}\right)\right\}\) in \(M_{2 \times 2}(\mathbb{R})\).
(b) \(\left\{x^3+2 x^2,-x^2+3 x+1, x^3-x^2+2 x-1\right\}\) in $P_3(\mathbb{R})\).
(13) Determine the subspace of the vector space \(P_2\) spanned by \(p_1(x)=1+3 x, p_2(x)=x+x^2\), and decide whether \(\left\{p_1, p_2\right\}\) is a spanning set for \(P_2\).
(14) Show that the set of solutions to the systems of linear equations
\( \begin{array}{r} x_1-2 x_2+x_3=0 \\ 2 x_1-3 x_2+x_3=0 \end{array} \)
is a subspace of \(\mathbb{R}^3\). Find a basis for this subspace.
(15) Determine whether the given set of vectors in \(\mathbb{R}^n\) is linearly dependent or linearly independent:
(a) \(\{(1,2,3),(1,0,1),(1,-1,5)\}\)
(b) \(\{(1,-1,3,-1),(1,-1,4,2),(1,-1,5,7)\}\)
(16) Let \(V\) be a vector space over \(F\). A subset \(\mathcal{B}\) of \(V\) is called a basis if it in linearly independent and it spans \(V\) over F. Prove that a set \(\left\{v_1, v_2, \ldots, v_n\right\}\) of vectors in a vector space \(V\) is a basis if and only if every \(v \in V\) can be written uniquely as \(v=c_1 v_1+c_2 v_2+\cdots+c_n v_n\) where \(c_i \in \mathbb{F}\) for all \(i\).
(17) Determine a basis for the subspace of \(\mathbb{R}^n\) spanned by the given set of vectors:
(a) \(\{(1,3,3),(-3,-9,-9),(1,5,-1),(2,7,4),(1,4,1)\}\)
(b) \(\{(1,1,-1,2),(2,1,3,-4),(1,2,-6,10)\}\)
(18) Show that the set vectors \(\left\{f_1, f_2, f_3\right\}\) is linearly independent where \(f_1(x)=2 x-3, f_2(x)=x^2+1, f_3(x)=2 x^2-x\). Complete the set to form a basis for \(P_3\), the set of all polynomials of degree no more than \(3\) .
(19) (a) Find a basis and the dimension of the subspace
\( W=\{(a+b+2 c, 2 a+2 b+4 c+d, b+c+d, 3 a+3 c+d): a, b, c, d \in \mathbb{R}\} . \)
(b) Let \(u, v\), and \(w\) be distinct vectors of a vector space \(V\). Show that if \(\{u, v, w\}\) is a basis of \(V\), then \(\{u+v+w, v+w, w\}\) is also a basis of \(V\).
(c) Find a basis of the space \(U=\left\{p \in P_4(\mathbb{R}): p(6)=0\right\}\).
(20) Prove that the real vector space of all continuous real-valued functions on the interval \([0,1]\) is infinite-dimensional.
(1) (a) No (b) Yes (c) No (d) Yes (e) Yes
(2) No
(3) (a) Yes (b) Yes (c) No (d) No
(4) \(\left\{\left(\begin{array}{ll}x & y \\ 0 & 0\end{array}\right): x, y \in \mathbb{R}\right\}\)
(5) Yes
(6)
(7) (a) \(V=\mathbb{R}^2, W=\{(x, 0): x \in \mathbb{R}\}, U_1=\{(0, y): y \in \mathbb{R}\}, U_2=\left\{(x, y) \in \mathbb{R}^2: x=y\right\}\) then \(V=U_1 \oplus W=U_2 \oplus W\) but \(U_1 \neq U_2\).
(8) (a) Yes (b) No (c) No
(9) No
(10) Yes
(11)
(12) (a) Linearly dependent (b) Linearly independent
(13) Not same
(14) \(\{(1,1,1)\}\)
(15) (a) Linearly dependent (b) Linearly independent
(17) (a) \(\{(1,3,3),(1,5,-1)\}\) (answers may vary)
(b) \(\{(1,1,-1,2),(2,1,3,-4)\}\) (answers may vary)
(18) \(\left\{f_1, f_2, f_3, f_4\right\}\) where \(f_4(x)=x^3\) (answers may vary)
(19) (a) \(\{(1,2,0,3),(1,2,1,0),(0,1,1,1)\}\) and dimension is 3
(b)
(c) \(\left\{(x-6), x(x-6), x^2(x-6), x^3(x-6)\right\}\).
(20)
(a) The set \[ \left\{(a, b) \in \mathbb{R}^2: b=5 a+1\right\}, \] with respect to usual addition and multiplication.
(b) The set \(M_n(\mathbb{R})\) of all \(n \times n\) real matrices over \(\mathbb{R}\) with respect to usual addition and multiplication.
(c) The set \(\mathbb{R}^2\) over \(\mathbb{R}\) with respect to the addition and scalar multiplication defined by \( \lambda \cdot(a, b):=(\lambda a, 0) \) for \((a, b) \in \mathbb{R}^2, \lambda \in \mathbb{R}\).
(d) The set \(C[a, b]\) of all real valued continuous functions defined on the interval \([a, b]\), with respect to the following addition and scalar multiplication: \((f+g)(t):=f(t)+g(t)\) and \((\lambda \cdot f)(t):=\lambda f(t)\) for \(f, g \in C[a, b], t \in[a, b], \lambda \in \mathbb{R}\).
(e) The set of all positive real numbers over \(\mathbb{R}\), with respect to the following addition and scalar multiplication: \(x+y:=x y\) and \(\lambda \cdot x:=x^\lambda\) for \(x, y \in \mathbb{R}, \lambda \in \mathbb{R}\).
(f) The set \(P_n\) of all real polynomials of order less than or, equal to \(n\), with respect to usual addition and scalar multiplication.
(2) Let \(V=\left\{\left(a_1, a_2\right): a_1, a_2 \in \mathbb{R}\right\}\). For \(\left(a_1, a_2\right),\left(b_1, b_2\right) \in V\) and \(c \in \mathbb{R}\), define
\( \left(a_1, a_2\right)+\left(b_1, b_2\right)=\left(a_1+2 b_1, a_2+3 b_2\right) \text { and } c\left(a_1, a_2\right)=\left(c a_1, c a_2\right) \text {. } \)
Is \(V\) a vector space over \(\mathbb{R}\) with these operations? Justify your answer.
(3) Determine whether the following sets are subspaces of \(R^3\) under the operations of addition and scalar multiplication defined on \mathbb{R}^3\). Justify your answers.
(a) \(W_1=\left\{\left(a_1, a_2, a_3\right) \in \mathbb{R}^3: a_1=3 a_2\right.\) and \(\left.a_3=-a_2\right\}\)
(b) \(W_2=\left\{\left(a_1, a_2, a_3\right) \in \mathbb{R}^3: 2 a_1-7 a_2+a_3=0\right\}\)
(c) \(W_3=\left\{\left(a_1, a_2, a_3\right) \in \mathbb{R}^3: a_1+2 a_2-3 a_3=1\right\}\)
(d) \(W_4=\left\{\left(a_1, a_2, a_3\right) \in \mathbb{R}^3: 5 a_1^2-3 a_2^2+6 a_3^2=0\right\}\)
(4) Describe the smallest subspace of the space \(M_2(\mathbb{R})\) containing \( \left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right)\) and \(\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)\). Justify your answer.
(5) Determine whether or not the set \(\left\{f \in C[0,1]: \int_0^1 f(x) d x=0\right\}\) is a subspace of \(C[0,1]\).
(6) Show that the subspaces of \(\mathbb{R}^3\) are precisely \(\{0\}, \mathbb{R}^3\), all lines in \(\mathbb{R}^3\) through the origin, and all planes in \(\mathbb{R}^3\) through the origin.
(7) If \(U_1, U_2\) are two subspaces of a vector space \(V\) then \(U_1+U_2\) is the direct sum \(U_1 \oplus U_2\) of \(U_1, U_2\) if every element \(u\) of \(U_1+U_2\) can be uniquely written as \(u=u_1+u_2\) where \(u_1 \in U_1\) and \(u_2 \in U_2\).
(a) Prove or give a counterexample: if \(U_1, U_2, W\) are subspaces of a vector space \(V\) such that \(V=U_1 \oplus W=U_2 \oplus W\) then \(U_1=U_2\).
(b) Let \(U_e\) denote the set of real-valued even functions on \(\mathbb{R}\) and let \(U_o\) denote the set of real-valued odd functions on \(\mathbb{R}\). Show that \(\mathbb{R}^{\mathbb{R}}=U_e \oplus U_o\), where \(\mathbb{R}^{\mathbb{R}}\) denotes the space of all functions from \(\mathbf{R}\) to \(\mathbf{R}\).
(8) For each of the following lists of vectors in \(\mathbb{R}^3\), determine whether the first vector can be expressed as a linear combination of the other two.
(a) \((-2,0,3),(1,3,0),(2,4,-1)\)
(b) \((3,4,1),(1,-2,1),(-2,-1,1)\)
(c) \((5,1,-5),(1,-2,-3),(-2,3,-4)\)
(9) Determine whether the vectors \(v_1=(1,-1,4), v_2=(-2,1,3)\) and \(v_3=(4,-3,5)\) span \(\mathbf{R}^3\).
(10) Verify that \(A_1=\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right), A_2=\left(\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right), A_3=\left(\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right)\), and \(A_4=\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right)\) span \(M_2(\mathbb{R})\).
(11) Show that if
\( M_1=\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right), \quad M_2=\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right), \quad \text { and } \quad M_3=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \)
then the span of \(\left\{M_1, M_2, M_3\right\}\) is the set of all \(2 \times 2\) symmetric matrices.
(12) Determine whether the following sets are linearly dependent or linearly independent.
(a) \(\left\{\left(\begin{array}{rr}1 & -3 \\ -2 & 4\end{array}\right),\left(\begin{array}{rr}2 & -1 \\ 3 & 2\end{array}\right),\left(\begin{array}{rr}1 & -8 \\ -9 & 10\end{array}\right)\right\}\) in \(M_{2 \times 2}(\mathbb{R})\).
(b) \(\left\{x^3+2 x^2,-x^2+3 x+1, x^3-x^2+2 x-1\right\}\) in $P_3(\mathbb{R})\).
(13) Determine the subspace of the vector space \(P_2\) spanned by \(p_1(x)=1+3 x, p_2(x)=x+x^2\), and decide whether \(\left\{p_1, p_2\right\}\) is a spanning set for \(P_2\).
(14) Show that the set of solutions to the systems of linear equations
\( \begin{array}{r} x_1-2 x_2+x_3=0 \\ 2 x_1-3 x_2+x_3=0 \end{array} \)
is a subspace of \(\mathbb{R}^3\). Find a basis for this subspace.
(15) Determine whether the given set of vectors in \(\mathbb{R}^n\) is linearly dependent or linearly independent:
(a) \(\{(1,2,3),(1,0,1),(1,-1,5)\}\)
(b) \(\{(1,-1,3,-1),(1,-1,4,2),(1,-1,5,7)\}\)
(16) Let \(V\) be a vector space over \(F\). A subset \(\mathcal{B}\) of \(V\) is called a basis if it in linearly independent and it spans \(V\) over F. Prove that a set \(\left\{v_1, v_2, \ldots, v_n\right\}\) of vectors in a vector space \(V\) is a basis if and only if every \(v \in V\) can be written uniquely as \(v=c_1 v_1+c_2 v_2+\cdots+c_n v_n\) where \(c_i \in \mathbb{F}\) for all \(i\).
(17) Determine a basis for the subspace of \(\mathbb{R}^n\) spanned by the given set of vectors:
(a) \(\{(1,3,3),(-3,-9,-9),(1,5,-1),(2,7,4),(1,4,1)\}\)
(b) \(\{(1,1,-1,2),(2,1,3,-4),(1,2,-6,10)\}\)
(18) Show that the set vectors \(\left\{f_1, f_2, f_3\right\}\) is linearly independent where \(f_1(x)=2 x-3, f_2(x)=x^2+1, f_3(x)=2 x^2-x\). Complete the set to form a basis for \(P_3\), the set of all polynomials of degree no more than \(3\) .
(19) (a) Find a basis and the dimension of the subspace
\( W=\{(a+b+2 c, 2 a+2 b+4 c+d, b+c+d, 3 a+3 c+d): a, b, c, d \in \mathbb{R}\} . \)
(b) Let \(u, v\), and \(w\) be distinct vectors of a vector space \(V\). Show that if \(\{u, v, w\}\) is a basis of \(V\), then \(\{u+v+w, v+w, w\}\) is also a basis of \(V\).
(c) Find a basis of the space \(U=\left\{p \in P_4(\mathbb{R}): p(6)=0\right\}\).
(20) Prove that the real vector space of all continuous real-valued functions on the interval \([0,1]\) is infinite-dimensional.
ANSWERS
(1) (a) No (b) Yes (c) No (d) Yes (e) Yes
(2) No
(3) (a) Yes (b) Yes (c) No (d) No
(4) \(\left\{\left(\begin{array}{ll}x & y \\ 0 & 0\end{array}\right): x, y \in \mathbb{R}\right\}\)
(5) Yes
(6)
(7) (a) \(V=\mathbb{R}^2, W=\{(x, 0): x \in \mathbb{R}\}, U_1=\{(0, y): y \in \mathbb{R}\}, U_2=\left\{(x, y) \in \mathbb{R}^2: x=y\right\}\) then \(V=U_1 \oplus W=U_2 \oplus W\) but \(U_1 \neq U_2\).
(8) (a) Yes (b) No (c) No
(9) No
(10) Yes
(11)
(12) (a) Linearly dependent (b) Linearly independent
(13) Not same
(14) \(\{(1,1,1)\}\)
(15) (a) Linearly dependent (b) Linearly independent
(17) (a) \(\{(1,3,3),(1,5,-1)\}\) (answers may vary)
(b) \(\{(1,1,-1,2),(2,1,3,-4)\}\) (answers may vary)
(18) \(\left\{f_1, f_2, f_3, f_4\right\}\) where \(f_4(x)=x^3\) (answers may vary)
(19) (a) \(\{(1,2,0,3),(1,2,1,0),(0,1,1,1)\}\) and dimension is 3
(b)
(c) \(\left\{(x-6), x(x-6), x^2(x-6), x^3(x-6)\right\}\).
(20)
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