Linear Transformations

Let $E$ and $F$ be vector spaces over a field $K$, and let $S : E \to F$ be a linear map.

Let $V$ be a finite-dimensional vector space and $T : V \to V$ a linear transformation with kernel of dimension $k$. Show that there is a basis $\alpha_{1}, \ldots , \alpha_{n}$ of $V$ such that $T\alpha_{k+1}, \ldots , T\alpha_{n}$ is a basis for the range of $T$.

Let $V$ and $W$ be finite-dimensional vector spaces over a field, and $T : V \to W$ a linear transformation. Let $X$ be a subspace of $W$. Prove that

Let $V$ be a finite-dimensional vector space, and let $A$ and $B$ be linear operators on $V$. Prove that

Let $A$ and $B$ be linear transformations of a finite-dimensional vector space $V$ into a vector space $W$ over a field $\mathbf{F}$, such that

Let $L$ be a linear operator on a finite-dimensional vector space $V$ over a field $\mathbf{F}$, and let $W \subset V$ be an $L$-invariant subspace. Let $L_{W}$ denote the restriction of $L$ to $W$. Prove that the characteristic polynomial of $L_{W}$ divides the characteristic polynomial of $L$.

Let $V$ be a finite-dimensional vector space over a field, $W$ a subspace, and $f$ an endomorphism of $V$ such that $f(V) \subset W \subset V$. Prove that the trace of $f$ equals the trace of $f|_{W}$, the restriction of $f$ to $W$.

Let $V$ be an $n$-dimensional vector space over a field $\mathbf{F}$, and let $T : V \to V$ be a linear transformation. Prove that $T$ has a nonzero invariant subspace of dimension, at most, $k$ if and only if the characteristic polynomial of $T$ has a factor of degree $k$ in $\mathbf{F}[t]$.

Let $T$ be a linear transformation of $\mathbb {R}^{n}$ into itself, and suppose $n > 1$. Prove that $\mathbb {R}^{n}$ has a $T$-invariant subspace $M$ of dimension $2$ (that is, $TM \subset M$).

Let $V$ be a finite-dimensional vector space over a field $\mathbf{F}$, and let $L : V \to V$ be a linear transformation with characteristic polynomial $\chi (t)$. Suppose $\chi (t) = \chi_{1}(t)\chi_{2}(t)$, where $\chi_{1}(t)$ and $\chi_{2}(t)$ are relatively prime elements of $\mathbf{F}[t]$. Let $V_{i} = \left\{ v \in V \mid \chi_{i}(L)(v) = 0\right\}$, for $i = 1, 2$. Show that $V$ is the direct sum of $V_{1}$ and $V_{2}$.

Let $T$ be the linear transformation of $\mathbb {Q}^{3}$ into itself given by

Show that if $x$ is any nonzero vector in $\mathbb {Q}^{3}$, then $x$, $y = Tx$, and $z = Ty$ are linearly independent, and, thus, form a basis of $\mathbb {Q}^{3}$.

Let $A$ and $B$ be linear transformations of a finite-dimensional real vector space $V$ such that $A^{2} = A$ and $B^{2} = B$. Suppose, in addition, that $1$ is not an eigenvalue of $AB$. Prove that $1$ is not an eigenvalue of $A+B$.

Let $V$ be a finite-dimensional real vector space, $B : V \times V \to \mathbb {R}$ a nondegenerate bilinear form, and $T : V \to V$ a linear map. Define $T^{*}$ by $B(Tv,w) = B(v,T^{*}w)$ for all $v, w \in V$. Show that $T$ and $T^{*}$ have the same eigenvalues.

Let $f_{1}, \ldots , f_{n}$ be linear functionals on the finite dimensional real vector space $\mathbb {R}^{m}$, and suppose that $f = (f_{1}, \ldots , f_{n}) : \mathbb {R}^{m} \to \mathbb {R}^{n}$ has rank $n-1$. Show that there is a nonzero point $v \in \mathbb {R}^{m}$ satisfying: for all $i$, either $f_{i}(v) = 0$ or there is a $j \neq i$ such that $f_{i}(v)$ and $f_{j}(v)$ have opposite (nonzero) signs.

Let $S$ be a finite subset of $\mathbb {R}^{n}$ containing $0$ and let $d(s,t)$ denote the usual Euclidean distance between points $s, t \in \mathbb {R}^{n}$. Suppose $\varphi : S \to \mathbb {R}^{n}$ satisfies $d(\varphi (s),\varphi (t)) = d(s,t)$ for all $s, t \in S$ and $\varphi (0) = 0$. Prove that $\varphi$ extends to a linear transformation of $\mathbb {R}^{n}$ into itself.

Let $T : \mathbb {R}^{2} \to \mathbb {R}^{2}$ be an isometry (that is, $\left\| T(x)-T(y)\right\| = \left\| x-y\right\|$ for all $x, y \in \mathbb {R}^{2}$). Prove that $T(x) = T(0)+U(x)$, where $U$ is an orthogonal linear transformation.

Let $V$ be an $n$-dimensional vector space of real-valued functions on a set $X$. Prove that there exist points $x_{1}, \ldots , x_{n}$ in $X$ and functions $f_{1}, \ldots , f_{n}$ in $V$ such that $\operatorname {det}\left((f_{i}(x_{j}))\right) \neq 0$.

Let $X$ be a set. Prove that for every finite-dimensional subspace $V$ of the vector space of continuous real valued functions on $X$, there exists a basis $f_{1}, \ldots , f_{n}$ for $V$ and points $x_{1}, \ldots , x_{n}$ in $X$ such that $f_{i}(x_{j}) = \delta_{ij}$.

Prove that for each positive integer $n$ there exist real numbers $c_{-n}, c_{-n+1}, \ldots , c_{n-1}, c_{n}$ such that

for every real polynomial $p(x)$ of degree, at most, $2n$.

Let $T$ be a linear operator on a finite-dimensional real vector space $V$. Suppose $T$ has $n$ distinct real eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$, where $n = \dim V$. Show that there is a basis of $V$ with respect to which the matrix of $T$ is upper-triangular.

Let $\mathbf{F}$ be a finite field. Let $V$ be a finite-dimensional vector space over $\mathbf{F}$. Prove that if $P : V \to V$ is an invertible linear transformation, then there is an integer $n \geq 1$ such that $P^{n} = I$, where $I$ denotes the identity transformation.

Let $A$ be a real $n \times n$ matrix such that $A^{3} = 0$. Prove that $I+A$ is invertible, and give a formula for its inverse.

Let $T$ be a linear operator on a $3$-dimensional real vector space $V$.

Let $A$ be a real symmetric orthogonal $n \times n$ matrix. Describe $A$ geometrically if $n = 3$.

Let $R$ and $S$ be rotations in $\mathbb {R}^{3}$ about two axes through the origin. Show that the composition $T = SR$ is a rotation.

Let $e = (a,b,c)$ be a unit vector in $\mathbb {R}^{3}$. Let $T$ be the linear transformation which reflects each vector across the line through the origin in the direction $e$. Find the matrix, in terms of $a$, $b$, and $c$, that represents $T$ with respect to the standard basis of $\mathbb {R}^{3}$.

Let $T$ be a linear transformation on $\mathbb {R}^{3}$ whose matrix, with respect to the standard basis, is orthogonal and has determinant $1$. Assume $T$ is not the identity transformation, but $T(1,1,1) = (1,1,1)$. Given that the eigenvalues of $T$ are $10$, $i$, $-i$, and that the minimal polynomial of $T$ has degree $3$, calculate the matrix of $T$ with respect to the standard basis.

Let $A$ be a $3 \times 3$ real orthogonal matrix with $\det A = 1$. (Such a matrix represents a rotation of $\mathbb {R}^{3}$.) Prove that $1$ is an eigenvalue of $A$. (This shows that every rotation of $\mathbb {R}^{3}$ has an axis; i.e., a line which is left fixed.)

Let $P$ be the vector space of all polynomials in one variable with real coefficients. Let $E : P \to P$ be the linear operator "differentiate and then multiply by $x$," i.e., $E(p) = xp'$. Prove that $E$ is not invertible.

Let $D$ be the derivative operator on the vector space $P$ of all polynomials in one variable with real coefficients. Given a polynomial $\pi (x)$, we obtain an operator $\pi (D)$ on $P$. Show that if $\pi (x)$ is a nonconstant polynomial and $a$ is a root of $\pi (x)$, then $(x-a)^{k}$ divides $\pi (x)$, where $k$ is minimal such that the operator $\pi (D)$ annihilates all polynomials of degree, at most, $k$.

Let $D$ denote the differentiation operator $d/dx$ on the (real) vector space $V$ of all polynomials of degree, at most, $10$ in the variable $x$.

Let $x_{0}, x_{1}, \ldots , x_{n}$ be distinct real numbers. Show that there exist real numbers $a_{0}, a_{1}, \ldots , a_{n}$ such that

for every polynomial $p(t)$ of degree, at most, $n$.

Let $I_{1}, \ldots , I_{n}$ be disjoint closed nonempty subintervals of $\mathbb {R}$.