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Linear Transformation Between Vector Spaces

Linear Transformation Between Vector Spaces

Definition

Let $V$ and $W$ be vector spaces over a scalar field $K$.

  1. A function $T:V \to W$ is called a linear transformation if $T$ satisfies the following two linearity conditions: For any $\mathbf{x}, \mathbf{y}\in V$ and $c\in K$, we have
    1. $T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y})$
    2. $T(c\mathbf{x})=cT(\mathbf{x})$
  2. The nullspace $\calN(T)$ of a linear transformation $T:V \to W$ is
    \[\calN(T)=\{\mathbf{x}\in V \mid T(\mathbf{x})=\mathbf{0}\}.\]
  3. The nullity of $T$ is the dimension of $\calN(T)$.
  4. The range $\calR(T)$ of a linear transformation $T:V \to W$ is
    \[\calR(T)=\{\mathbf{y}\in W \mid \mathbf{y}=T(\mathbf{x}) \text{ for some } \mathbf{x}\in V\}.\]
  5. The rank of $T$ is the dimension of $\calR(T)$.
Summary

Let $T:V\to W$ be a linear transformation.

  1. $T$ maps the zero vector $\mathbf{0}_V$ of $V$ to the zero vector $\mathbf{0}_W$ of $W$. That is, $T(\mathbf{0}_V)=\mathbf{0}_W$.

=solution

Problems

  1. Let $C (\mathbb{R})$ be the vector space of real functions. Define the map $T$ by $T(f)(x) = (f(x))^2$ for $f \in C(\mathbb{R})$. Determine if $T$ is a linear transformation or not. If it is, determine the range of $T$.
  2. For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, for $f \in \mathrm{P}_n$, $T (f) (x) = x f(x)$. Prove that $T$ is a linear transformation, and find its range and nullspace.
  3. Let $C ([0, 3] )$ be the vector space of real functions on the interval $[0, 3]$. Let $\mathrm{P}_3$ denote the set of real polynomials of degree $3$ or less. Define the map $T : C ([0, 3] ) \rightarrow \mathrm{P}_3 $ by
    \[T(f)(x) = f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) \cdot x^3.\] Determine if $T$ is a linear transformation. If it is, determine its nullspace.

  4. Let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_4 \rightarrow \mathrm{P}_{4}$ be the map defined by, for $f \in \mathrm{P}_4$, $T (f) (x) = f(x) – x – 1$. Determine if $T(x)$ is a linear transformation. If it is, find the matrix representation of $T$ relative to the standard basis of $\mathrm{P}_4$.
  5. Let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_3 \rightarrow \mathrm{P}_{5}$ be the map defined by, for $f \in \mathrm{P}_3$, $T(f) (x) = ( x^2 – 2) f(x)$. Determine if $T(x)$ is a linear transformation. If it is, find the matrix representation of $T$ relative to the standard basis of $\mathrm{P}_3$ and $\mathrm{P}_{5}$.
  6. Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients.
    (a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by
    \[T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)\] for any $f(x)\in P_3$ is a linear transformation.
    (b) Let $B=\{1, x, x^2, x^3\}$ be a basis of $P_3$. With respect to the basis $B$, find the matrix representation of the linear transformation $T$ in part (a).

  7. For an integer $n > 0$, let $\mathrm{P}_n$ denote the vector space of polynomials with real coefficients of degree $2$ or less. Define the map $T : \mathrm{P}_2 \rightarrow \mathrm{P}_4$ by
    \[ T(f)(x) = f(x^2).\] Determine if $T$ is a linear transformation. If it is, find the matrix representation for $T$ relative to the basis $\mathcal{B} = \{ 1 , x , x^2 \}$ of $\mathrm{P}_2$ and $\mathcal{C} = \{ 1 , x , x^2 , x^3 , x^4 \}$ of $\mathrm{P}_4$.

  8. Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less. Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$. For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation of $T$ with respect to the basis $B$. For $f(x)\in P_2(\R)$, define $T$ as follows.
    (a) \[T(f(x))=\frac{\mathrm{d}^2}{\mathrm{d}x^2} f(x)-3\frac{\mathrm{d}}{\mathrm{d}x}f(x)\] (b) \[T(f(x))=\int_{-1}^1\! (t-x)^2f(t) \,\mathrm{d}t\] (c) \[T(f(x))=e^x \frac{\mathrm{d}}{\mathrm{d}x}(e^{-x}f(x))\]
  9. Let $U$ and $V$ be vector spaces over a scalar field $\F$. Define the map $T:U\to V$ by $T(\mathbf{u})=\mathbf{0}_V$ for each vector $\mathbf{u}\in U$.
    (a) Prove that $T:U\to V$ is a linear transformation. (Hence, $T$ is called the zero transformation.)
    (b) Determine the null space $\calN(T)$ and the range $\calR(T)$ of $T$.

  10. Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form $x+y \sqrt{2}$ where $x,y \in \Q$. You may take for granted that the set $V$ is a vector space over the field $\Q$.
    (a) Show that $B=\{1, \sqrt{2}\}$ is a basis for the vector space $V$ over $\Q$.
    (b) Let $\alpha=a+b\sqrt{2} \in V$, and let $T_{\alpha}: V \to V$ be the map defined by
    \[ T_{\alpha}(x+y\sqrt{2}):=(ax+2by)+(ay+bx)\sqrt{2}\in V\] for any $x+y\sqrt{2} \in V$. Show that $T_{\alpha}$ is a linear transformation.
    (c) Let $\begin{bmatrix}
    x \\
    y
    \end{bmatrix}_B=x+y \sqrt{2}$. Find the matrix $T_B$ such that
    \[ T_{\alpha} (x+y \sqrt{2})=\left( T_B\begin{bmatrix}
    x \\
    y
    \end{bmatrix}\right)_B,\] and compute $\det T_B$.
    (The Ohio State University)

  11. Let $V$ be the vector space of all $2\times 2$ real matrices and let $P_3$ be the vector space of all polynomials of degree $3$ or less with real coefficients. Let $T: P_3 \to V$ be the linear transformation defined by
    \[T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix}
    a_0+a_2 & -a_0+a_3\\
    a_1-a_2 & -a_1-a_3
    \end{bmatrix}\] for any polynomial $a_0+a_1x+a_2x^2+a_3 \in P_3$. Find a basis for the range of $T$, $\calR(T)$, and determine the rank of $T$, $\rk(T)$, and the nullity of $T$, $\nullity(T)$. Also, prove that $T$ is not injective.

  12. Let $V$ be the vector space of $2 \times 2$ matrices with real entries, and $\mathrm{P}_3$ the vector space of real polynomials of degree 3 or less. Define the linear transformation $T : V \rightarrow \mathrm{P}_3$ by
    \[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = 2a + (b-d)x – (a+c)x^2 + (a+b-c-d)x^3.\] Find the rank and nullity of $T$.

  13. Let $V$ denote the vector space of $2 \times 2$ matrices, and $W$ the vector space of $3 \times 2$ matrices. Define the linear transformation $T : V \rightarrow W$ by
    \[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = \begin{bmatrix} a+b & 2d \\ 2b – d & -3c \\ 2b – c & -3a \end{bmatrix}.\] Find a basis for the range of $T$.

  14. Let $C([-1, 1])$ denote the vector space of real-valued functions on the interval $[-1, 1]$. Define the vector subspace
    \[W = \{ f \in C([-1, 1]) \mid f(0) = 0 \}.\] Define the map $T : C([-1, 1]) \rightarrow W$ by $T(f)(x) = f(x) – f(0)$. Determine if $T$ is a linear map. If it is, determine its nullspace and range.

  15. Let $\mathbb{R}^2$ be the vector space of size-2 column vectors. This vector space has an inner product defined by $ \langle \mathbf{v} , \mathbf{w} \rangle = \mathbf{v}^\trans \mathbf{w}$. A linear transformation $T : \R^2 \rightarrow \R^2$ is called an orthogonal transformation if for all $\mathbf{v} , \mathbf{w} \in \R^2$, $\langle T(\mathbf{v}) , T(\mathbf{w}) \rangle = \langle \mathbf{v} , \mathbf{w} \rangle$. For a fixed angle $\theta \in [0, 2 \pi )$ , define the matrix $[T] = \begin{bmatrix} \cos (\theta) & – \sin ( \theta ) \\ \sin ( \theta ) & \cos ( \theta ) \end{bmatrix}$ and the linear transformation $T : \R^2 \rightarrow \R^2$ by $T( \mathbf{v} ) = [T] \mathbf{v}$. Prove that $T$ is an orthogonal transformation.
  16. Let $V$ be a real vector space of all real sequences
    \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ for $k=1, 2, \dots$.
    (a) Let
    \begin{align*}
    \mathbf{u}_1&=(1, 0, -3, -15, -66, \dots)\\
    \mathbf{u}_2&=(0, 1, 5, 22, 95, \dots)
    \end{align*}
    be vectors in $U$. Prove that $\{\mathbf{u}_1, \mathbf{u}_2\}$ is a basis of $U$ and conclude that the dimension of $U$ is $2$.
    (b) Let $T$ be a map from $U$ to $U$ defined by
    \[T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots). \] Verify that the map $T$ actually sends a vector $(a_i)_{i=1}^{\infty}\in V$ to a vector $T\big((a_i)_{i=1}^{\infty}\big)$ in $U$, and show that $T$ is a linear transformation from $U$ to $U$.
    (c) With respect to the basis $\{\mathbf{u}_1, \mathbf{u}_2\}$ obtained in (a), find the matrix representation $A$ of the linear transformation $T:U \to U$ from (b).

  17. Let $V$ be the vector space of all $n\times n$ real matrices. Let us fix a matrix $A\in V$. Define a map $T: V\to V$ by $T(X)=AX-XA$ for each $X\in V$.
    (a) Prove that $T:V\to V$ is a linear transformation.
    (b) Let $B$ be a basis of $V$. Let $P$ be the matrix representation of $T$ with respect to $B$. Find the determinant of $P$.

  18. The space $C^{\infty} (\mathbb{R})$ is the vector space of real functions which are infinitely differentiable. Let $T : C^{\infty} (\mathbb{R}) \rightarrow \mathrm{P}_3$ be the map which takes $f \in C^{\infty}(\mathbb{R})$ to its third order Taylor polynomial, specifically defined by
    \[ T(f)(x) = f(0) + f'(0) x + \frac{f^{\prime\prime}(0)}{2} x^2 + \frac{f^{\prime \prime \prime}(0)}{6} x^3.\] Here, $f’, f^{\prime\prime}$ and $f^{\prime \prime \prime}$ denote the first, second, and third derivatives of $f$, respectively. Prove that $T$ is a linear transformation.

  19. Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less. Consider the differentiation linear transformation $T: P_n\to P_n$ defined by
    \[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\] (a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a basis of $P_2$. Find the matrix representation $A$ of the linear transformation $T$ with respect to the basis $B$.
    (b) Compute $A^3$, where $A$ is the matrix obtained in part (a).
    (c) If you computed $A^3$ in part (b) directly, then is there any theoretical explanation of your result?
    (d) Now we consider the general case. Let $B$ be any basis of the vector space of $P_n$ and let $A$ be the matrix representation of the linear transformation $T$ with respect to the basis $B$. Prove that without any calculation that the matrix $A$ is nilpotent.

  20. Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation). Let $V$ be the vector space of all linear transformations from $W$ to $W$. The addition and the scalar multiplication of $V$ are given by those of linear transformations. Let $T_1, T_2, T_3$ be the elements in $V$ defined by
    \begin{align*}
    T_1\left(\, f(x) \,\right)&=\frac{\mathrm{d}}{\mathrm{d}x}f(x)\\[6pt] T_2\left(\, f(x) \,\right)&=\frac{\mathrm{d}^2}{\mathrm{d}x^2}f(x)\\[6pt] T_3\left(\, f(x) \,\right)&=\int_{0}^x \! f(t)\,\mathrm{d}t.
    \end{align*}
    Then determine whether the set $\{T_1, T_2, T_3\}$ are linearly independent or linearly dependent.

  21. Let $V$ denote the vector space of all real $2\times 2$ matrices. Suppose that the linear transformation from $V$ to $V$ is given as below.
    \[T(A)=\begin{bmatrix}
    2 & 3\\
    5 & 7
    \end{bmatrix}A-A\begin{bmatrix}
    2 & 3\\
    5 & 7
    \end{bmatrix}.\] Prove or disprove that the linear transformation $T:V\to V$ is an isomorphism.

  22. Let $U$ and $V$ be vector spaces over a scalar field $\F$. Let $T: U \to V$ be a linear transformation. Prove that $T$ is injective (one-to-one) if and only if the nullity of $T$ is zero.
  23. Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$. Consider a linear transformation $T:U\to V$. Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).
  24. Let $V$ be a vector space over the field of real numbers $\R$. Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$.
  25. For a set $S$ and a vector space $V$ over a scalar field $\K$, define the set of all functions from $S$ to $V$
    \[ \Fun ( S , V ) = \{ f : S \rightarrow V \} . \] For $f, g \in \Fun(S, V)$, $z \in \K$, addition and scalar multiplication can be defined by
    \[ (f+g)(s) = f(s) + g(s) \, \mbox{ and } (cf)(s) = c (f(s)) \, \mbox{ for all } s \in S . \] (a) Prove that $\Fun(S, V)$ is a vector space over $\K$. What is the zero element?
    (b) Let $S_1 = \{ s \}$ be a set consisting of one element. Find an isomorphism between $\Fun(S_1 , V)$ and $V$ itself. Prove that the map you find is actually a linear isomorpism.
    (c) Suppose that $B = \{ e_1 , e_2 , \cdots , e_n \}$ is a basis of $V$. Use $B$ to construct a basis of $\Fun(S_1 , V)$.
    (d) Let $S = \{ s_1 , s_2 , \cdots , s_m \}$. Construct a linear isomorphism between $\Fun(S, V)$ and the vector space of $n$-tuples of $V$, defined as
    \[ V^m = \{ (v_1 , v_2 , \cdots , v_m ) \mid v_i \in V \mbox{ for all } 1 \leq i \leq m \} . \] (e) Use the basis $B$ of $V$ to constract a basis of $\Fun(S, V)$ for an arbitrary finite set $S$. What is the dimension of $\Fun(S, V)$?
    (f) Let $W \subseteq V$ be a subspace. Prove that $\Fun(S, W)$ is a subspace of $\Fun(S, V)$.