Mathematical Immaturity

2.12 Exercises

$\quad$ In all exercises involving the vector space $V_n,$ the usual basis of unit coordinate vectors is to be chosen, unless another basis is specifically mentioned. In exercises concerned with the matrix of a linear transformation $T: V \rightarrow W$ where $V = W,$ we take the same basis in both $V$ and $W$ unless another choice is indicated.

• Determine the matrix of each of the following linear transformations of $V_n$ into $V_n:$
  (a) $\quad$ the identity transformation,
  (b) $\quad$ the zero transformation,
  (c) $\quad$ multiplication by a fixed scalar $c.$
• Determine the matrix for each of the following projections.
  (a) $\quad T: V_3 \rightarrow V_2,$ where $T(x_1, x_2, x_3) = (x_1, x_2).$
  (b) $\quad T: V_3 \rightarrow V_2,$ where $T(x_1, x_2, x_3) = (x_2, x_3).$
  (c) $\quad T: V_5 \rightarrow V_3,$ where $T(x_1, x_2, x_3, x_4, x_5) = (x_2, x_3, x_4).$
• A linear transformation $T: V_2 \rightarrow V_2$ maps the basis vectors $\mathbf{i}$ and $\mathbf{j}$ as follows: \begin{align*} T(\mathbf{i}) &= \mathbf{i} + \mathbf{j}, \quad T(\mathbf{j}) = \mathbf{2i} - \mathbf{j} \end{align*} (a) $\quad$ Compute $T(3\mathbf{i} - 4\mathbf{j})$ and $T^2(3\mathbf{i} - 4\mathbf{j})$ in terms of $\mathbf{i}$ and $\mathbf{j}.$
  (b) $\quad$ Determine the matrix of $T$ and of $T^2$.
  (c) $\quad$ Solve part (b) if the basis $(\mathbf{i}, \mathbf{j})$ is replaced by $(e_1, e_2),$ where $e_1 = \mathbf{i} - \mathbf{j},$ $e_2 = 3\mathbf{i} + \mathbf{j}.$
• A linear transformation $T: V_2 \rightarrow V_2$ is defined as follows: Each vector $(x, y)$ is reflected in the $y$-axis and then doubled in length to yield $T(x, y).$ Determine the matrix of $T$ and of $T^2.$
• Let $T: V_3 \rightarrow V_3$ be a linear transformation such that \begin{align*} T(\mathbf{k}) &= 2\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}, \quad T(\mathbf{j} + \mathbf{k}) = \mathbf{i}, \quad T(\mathbf{i} + \mathbf{j} + \mathbf{k}) = \mathbf{j} - \mathbf{k}. \end{align*} (a) $\quad$ Compute $T(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$ and determine the nullity and rank of $T.$
  (b) $\quad$ Determine the matrix of $T.$
• For the linear transformation in Exercise 5, choose both bases to be $(e_1, e_2, e_3),$ where $e_1 = (2, 3, 5),$ $e_2 = (1, 0, 0),$ $e_3 = (0, 1, -1),$ and determine the matrix of $T$ relative to the new bases.
• A linear transformation $T: V_3 \rightarrow V_2$ maps the basis vectors as follows: \begin{align*} T(\mathbf{i}) &= (0, 0), \quad T(\mathbf{j}) = (1, 1), \quad T(\mathbf{k}) = (1, -1) \end{align*} (a) $\quad$ Compute $T(4\mathbf{i} - \mathbf{j} + \mathbf{k})$ and determine the nullity and rank of $T.$
  (b) $\quad$ Determine the matrix of $T.$
  (c) $\quad$ Use the basis $(\mathbf{i}, \mathbf{j}, \mathbf{k})$ in $V_3$ and the basis $(w_1, w_2)$ in $V_2$ where $w_1 = (1, 1),$ $w_2 = (1, 2).$ Determine the matrix of $T$ relative to these bases
  (d) $\quad$ Find bases $(e_1, e_2, e_3)$ for $V_2$ and $(w_1, w_2)$ for $V_2$ relative to which the matrix of $T$ will be in diagonal form.
• A linear transformation $T: V_2 \rightarrow V_3$ maps the basis vectors as follows: \begin{align*} T(\mathbf{i}) = (1, 0, 1), \quad T(\mathbf{j}) = (-1, 0, 1). \end{align*} (a) $\quad$ Compute $T(2\mathbf{i} - 3\mathbf{j})$ and determine the nullity and rank of $T.$
  (b) $\quad$ Determine the matrix of $T.$
  (c) $\quad$ Find bases $(e_1, e_2)$ for $V_2$ and $(w_1, w_2, w_3)$ for $V_3$ relative to which the matrix of $T$ will be in diagonal form.
• Solve Exercise 8 if $T(\mathbf{i}) = (1, 0, 1)$ and $T(\mathbf{j}) = (1, 1, 1).$
• Let $V$ and $W$ be linear spaces, each with dimension 2 and each with basis $(e_1, e_2).$ Let $T: V \to W$ be a linear transformation such that \begin{align*} T(e_1 + e_2) &= 3e_1 + 9e_2, \qquad T(3e_1 + 2e_2) = 7e_1 + 23e_2. \end{align*} (a) $\quad$ Compute $T(e_2 - e_1)$ and determine the nullity and rank of $T.$
  (b) $\quad$ Determine the matrix of $T$ relative to the given basis of $T.$
  (c) $\quad$ Use the basis $(e_1, e_2)$ for $V$ and find a new basis of the form $(e_1 + ae_2, 2e_1 + be_2)$ for $W,$ relative to which the matrix of $T$ will be in diagonal form.

$\quad$ In the linear space of all real-valued functions, each of the following sets is independent and spans a finite-dimensional subspace $V.$ Use the given set as a basis for $V$ and let $D: V \rightarrow V$ be the differentiation operator. In each case, find the matrix of $D$ and of $D^2$ relative to this choice of basis.

• $(\sin x, \cos x).$
• $(1, x, e^x).$
• $(1, 1 + x, 1 + x + e^x).$
• $(e^x, x e^x).$
• $(-\cos x, \sin x).$
• $(\sin x, \cos x, x \sin x, x \cos x).$
• $(e^x \sin x, e^x \cos x).$
• $(e^{2x} \sin 3x, e^{2x} \cos 3x).$
• Choose the basis $(1, x, x^2, x^3)$ in the linear space $V$ of all real polynomials of degree $\leq 3.$ Let $D$ denote the differentiation operator and let $T: V \rightarrow V$ be the linear transformation which maps $p(x)$ onto $x p'(x).$ Relative to the given basis, determine the matrix of each of the following transformations: (a) $T;$ (b) $DT;$ (c) $TD;$ (d) $TD - DT;$ (e) $T^2;$ (f) $T^2 D^2 - D^2 T^2.$
• Refer to Exercise 19. Let $W$ be the image of $V$ under $TD.$ Find bases for $V$ and for $W$ relative to which the matrix of $TD$ is in diagonal form.