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 $\calR(T)$.

=solution

Problems

  1. 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).

  2. 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) \[T(f(x))=e^x \frac{\mathrm{d}}{\mathrm{d}x}(e^{-x}f(x))\]
  3. 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 $T$.

  4. 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)

  5. 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.

  6. 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).

  7. 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 $P$.

  8. 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.

  9. 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.

  10. 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.

  11. 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.
  12. 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).
  13. 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 $\R^n$.