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Linear Algebra with SageMath: A Gentle Introduction

Ajit Kumar

Section 4.7 Exercise Set

  1. Check if the following set of vectors are linearly independent or dependent.
    (i) \(\{(1,0,1,2), (0,1,1,2),(1,1,1,3)\}\)
    (ii) \(\left\{\begin{bmatrix}1 \amp 0 \\3 \amp 2 \end{bmatrix} , \begin{bmatrix}-1 \amp 2 \\3 \amp 2 \end{bmatrix} , \begin{bmatrix}5 \amp -6 \\-3 \amp -2 \end{bmatrix} \right\}\text{.}\)
    (iii) \(\{(1,0,3),(1,2,4),(1,4,5)\}\text{.}\)
    (iv) \(\{(1, 0, -2, 5), (2, 1, 0, -1), (1, 1, 2, 1)\}\text{.}\) (v) \(\{(1, 1, 0, 0), (1, 0, 1, 0), (0, 0, 1, 1), (0, 1, 0, 1)\}\)
  2. Show that
    (i) \(\{(1,-1),(2,1)\}\) is a basis of \(\R^2\text{.}\)
    (ii) \(\{(1,1,-1),(-1,1,1),(1,-1,1)\}\) is a basis of \(\R^3\text{.}\)
    (iii) \(\{(1,-1,1,2),(-1,1,1,2),(1,-1,2,1),(-1,1,2,1)\}\) is a basis of \(\R^4\text{.}\)
    (iv) Show that any \(n+1\) vectors in \(\R^n\) are linearly independent. State the result clearly that is used.
  3. Consider the plane \(W=\{(x_1,x_2,x_3)\in \R^3:x_1-2x_2+x_3=0\}\text{.}\) Find a basis of \(W\) and hence find the dimension of \(W\text{.}\)
  4. Find the dimensions of the following subspaces. (i) \(W:=\{(x_1,\ldots,x_n):x_1+\cdots+x_n=0\}\) and (ii) \(W=\{(x_1,x_2,x_3,x_4)\in \R^4:x_1=x_3,x_2=x_4\}\text{.}\)
  5. Consider bases \(\beta=\{(1,-1),(1,2)\}\) and \(\gamma =\{(2,3),(1,3)\}\) of \(\R^2\text{.}\) Let \(x=(5,7)\text{.}\) Find the coordinates \(x_\beta\) and \(x_\gamma\) of \(x\) with respect to \(\beta\) and \(\gamma\) respectively. Also find the matrix of change of basis \([I]_\beta^\gamma\text{.}\) Hence show that \(x_\gamma = [I]_\beta^\gamma x_\beta\text{.}\)
  6. Consider bases \(\beta=\{(1,-1,1),(1,1,-1),(-1,1,1)\}\) and \(\gamma =\{(1,2,3),(1,3,2),(2,3,1)\}\) of \(\R^3\text{.}\) Let \(x=(5,7,11)\text{.}\) Find the coordinates \(x_\beta\) and \(x_\gamma\) of \(x\) with respect to \(\beta\) and \(\gamma\) respectively. Also find the matrix of change of basis \([I]_\beta^\gamma\text{.}\) Hence show that \(x_\gamma = [I]_\beta^\gamma x_\beta\text{.}\)
  7. Consider a linear map \(T\colon \R^3\to \R^2\) defined by \(T\left(\begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix} \right)=\begin{bmatrix}2x_1-x_2+x_3\\x_1+x_2-x_3 \end{bmatrix}\text{.}\) Let us consider a basis \(\beta =\{v_1=(1,1,-1),v_2=(1,-1,1),v_3=(-1,1,1)\}\) of the domain and the standard basis \(\gamma=\{(1,-1),(1,1)\}\) on the codomain. Find the matrix of \(T\) with respect to the basis \(\beta\) and \(\gamma\text{.}\)
  8. Let \(T\colon \R^4\to \R^3\) and \(S\colon \R^3\to \R^4\) defined by
    \begin{equation*} T\left(\begin{bmatrix}x_1\\x_2\\x_3\\x_4 \end{bmatrix} \right):= \begin{bmatrix}x_{1} + x_{3} + x_{4} \\ x_{1} + x_{2} + 2 x_{3} - x_{4} \\ 2 x_{1} + x_{2} + 3 x_{3} - 2 x_{4} \end{bmatrix} \end{equation*}
    and
    \begin{equation*} S\left(\begin{bmatrix}y_1\\y_2\\y_3 \end{bmatrix} \right):= \begin{bmatrix}y_{1} + y_{3} \\ y_{1} + 3 y_{2} + 2 y_{3} \\ 2 y_{1} - y_{2} + 3 y_{3} \\ y_{2} - y_{3} \end{bmatrix} \end{equation*}
    Find the composition \(S\circ T\text{.}\) Find th matrix \(A\) of \(T\text{,}\) \(B\) of \(S\) and \(C\) of \(S\circ T\) with respect to the standard bases. Show that \(C=BA\text{.}\)
  9. Let \(T,S\colon \R^n\to \R^m\) be two linear maps. Then show that \(T+S\) is a linear map. Furthermore, the matrix of \(T+S\) is the sum of matrices of \(T\) and \(S\text{.}\)
  10. For the following linear transformation \(T\colon \R^2\to \R^2\text{.}\) Show that \(T\) is induced by a matrix and hence find the matrix. (i) \(T\) is reflection about \(y\) axis. (ii) \(T\) is reflection about the line \(y=x\) (iii) \(T\) is reflection about the line \(y=-x\) (iv) \(T\) is a clockwise rotation by an angle \(\pi/2\text{.}\)
  11. (i) Let \(T\colon \R^3\to \R^3\) be a linear transformation which is reflection about the \(xy\) plane. Write \(T\) explicitly and hence show that it is induced by a matrix. (ii) Let \(T\colon \R^3\to \R^3\) be a linear transformation which is reflection about the \(yz\) plane. Write \(T\) explicitly and hence show that it is induced by a matrix.
  12. Let \(W_1\) is a set of all \(n\times n\) real symmetric matrices and \(W_2\text{,}\) the set of all \(n\times n\) real skew-symmetric matrices. Then what is \(W_1+W_2\text{?}\) Justify your answer.
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