Group Theory and Linear algebra

Semester II 2011

Last updates: 14 August 2011

(1) Week 4: Vocabulary

(2) Week 4: Results

(3) Week 4: Examples and computations

Define eigenvalue, eigenvector and eigenspace and give some illustrative examples. | |

Define generalised eigenspace and give some illustrative examples. | |

Define $f$-invariant subspace and restriction of $f$ and give some illustrative examples. | |

Define complement (to a subspace) and give some illustrative examples. | |

Define monic polynomial and give some illustrative examples. | |

Define minimal polynomial and characteristic polynnomial and give some illustrative examples. | |

Define invertible matrix and give some illustrative examples. | |

Define define diagonal matrix, upper triangular matrix, strictly upper triangular matrix, and unipotent upper triangular matrix and give some illustrative examples. | |

Let $\mathbb{F}$ be a field and let $d,a\in \mathbb{F}\left[t\right]$. Define the ideal generated by $d$ and "$d$ divides $a$" give some illustrative examples. | |

Let $\mathbb{F}$ be a field and let $x,m\in \mathbb{F}\left[t\right]$. Define the greatest common divisor of $x$ and $m$ and give some illustrative examples. | |

Let $\mathbb{F}$ be a field and let $p\in \mathbb{F}\left[t\right]$. Define the degree of $p$ and monic polynomial and give some illustrative examples. |

Let $\mathbb{F}$ be a field and let $a,b\in \mathbb{F}\left[t\right]$. Show that there exist
$q,r\in \mathbb{F}\left[t\right]$
such that
- (a) $a=bq+r$,
- (b) Either $r=0$ or $\mathrm{deg}\left(r\right)<\mathrm{deg}\left(b\right)$
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Let $a,b\in \mathbb{Z}$. Show that there exist
$q,r\in \mathbb{Z}$
such that
- (a) $a=bq+r$, and
- (b) $0<r<\left|b\right|$.
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Let $a,b\in \mathbb{F}\left[t\right]$ and let
$d=\mathrm{gcd}(a,b)$.
Show that
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Let $a,b\in \mathbb{Z}$ and let
$d=\mathrm{gcd}(a,b)$.
Show that
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Let $f:V\to V$ be a linear transformation and let $W$ be an $f$-invariant subspace with $\mathrm{dim}\left(V\right)=n$ and $\mathrm{dim}\left(W\right)=m$. Let ${\mathcal{B}}_{1}=\{{w}_{1},\dots ,{w}_{m}\}$ be a basis for $W$ and extend it to a basis Let $\mathcal{B}=\{{w}_{1},\dots ,{w}_{m},{w}_{m+1},\dots ,{w}_{n}\}$ for $V$. Show that the matrix of $f$ with respect to $\mathcal{B}$ is of the block form $$\left(\begin{array}{cc}A& B\\ 0& D\end{array}\right)$$ where $A,B,D$ are matrices and $A$ is the $m\times m$ matrix of ${f}_{W}$ with respect to the basis ${\mathcal{B}}_{1}$. | |

Let $V$ be a finite dimensional vector space and let $U$
and $W$ be subspaces of $V$. Show that the following
are equivalent.
- (1) $U$ is a complement of $W$,
- (2) There is a basis $\mathcal{B}$ of $V$ of the form $\mathcal{B}={\mathcal{B}}_{1}\cup {\mathcal{B}}_{2}$, where ${\mathcal{B}}_{1}$ is a basis of $U$, ${\mathcal{B}}_{2}$ is a basis of $W$ and ${\mathcal{B}}_{1}\cap {\mathcal{B}}_{2}=\varnothing $,
- (3) $U\cap W=\left\{0\right\}$ and $\mathrm{dim}U=\mathrm{dim}V-\mathrm{dim}W$,
- (4) $V=U+W$ and $\mathrm{dim}U=\mathrm{dim}V-\mathrm{dim}W$.
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Let $V$ be a vector space and let $f$ be a linear transformation on $V$. Let $U$ and $W$ be complementary subspaces of $V$. Suppose that both $U$ and $W$ are $f$-invariant. Choose an ordered basis $\mathcal{B}$ of $V$ of the form $\mathcal{B}=({\mathcal{B}}_{1},{\mathcal{B}}_{2})$ where ${\mathcal{B}}_{1}$ is a basis of $U$ and ${\mathcal{B}}_{2}$ is a basis of $W$. Show that the matrix of $f$ with respect to $\mathcal{B}$ is of the "block diagonal" form: $$\left(\begin{array}{cc}A& 0\\ 0& D\end{array}\right)$$ where $A$ is the matrix of ${f}_{U}$ and $D$ is the matrix of ${f}_{W}$. | |

Let $\mathbb{F}$ be a field and let $V$ be a vector space over $\mathbb{F}$. Let $f:V\to V$ be a linear transformation and let $m\left(x\right)$ be the minimal polynomial of $f$. Show that if $q\left(x\right)$ is a polynomial with coefficients in $\mathbb{F}$ such that $q\left(f\right)=0$ then $m\left(x\right)$ divides $q\left(x\right)$. | |

Let $\mathbb{F}$ be a field and let $V$ be a vector space over $\mathbb{F}$. Let $f:V\to V$ be a linear transformation and let $m\left(x\right)$ be the minimal polynomial of $f$. Show that the roots of $m\left(x\right)$ are exactly the eigenvalues of $f$. | |

Let $\mathbb{F}$ be a field and let $V$ be a vector space over $\mathbb{F}$. Let $f:V\to V$ be a linear transformation and let $p\left(x\right)\in \mathbb{F}\left[x\right]$. Show that the null space of $p\left(f\right)$ is an $f$-invariant subspace of $V$. | |

Let $\mathbb{F}$ be a field and let $V$ be a vector space over $\mathbb{F}$. Let $f:V\to V$ be a linear transformation and let $m\left(x\right)$ be the minimal polynomial of $f$. Suppose that $m\left(x\right)$ can be factored as $m\left(x\right)=p\left(x\right)q\left(x\right)$ where $p\left(x\right)$ and $q\left(x\right)$ are polynomials with coefficients in $\mathbb{F}$ which have no common factor. Show that $V$ is a direct sum of $f$-invariant subspaces $$V={W}_{p}\oplus {W}_{q},$$ where ${W}_{p}$ and ${W}_{q}$ are the nulspaces of $p\left(f\right)$ and $q\left(f\right)$, respectively. Show that the restrictions ${f}_{{W}_{p}}$ and ${f}_{{W}_{q}}$ have minimal polynomials $p\left(x\right)$ and $q\left(x\right)$, respectively. | |

Let $\mathbb{F}$ be a field and let $V$ be a vector space over $\mathbb{F}$. Let $f:V\to V$ be a linear transformation and let $m\left(x\right)$ be the minimal polynomial of $f$. Suppose that $m\left(x\right)={q}_{1}\left(x\right){q}_{2}\left(x\right)\cdots {q}_{k}\left(x\right)$ where ${q}_{i}\left(x\right)$ has no common factor with ${q}_{j}\left(x\right)$ if $i\ne j$. Let ${W}_{i}$ be the nullspace of ${q}_{i}\left(f\right)$. Suppose that ${\mathcal{B}}_{i}$ is an ordered basis of ${W}_{i}$. Show that $\mathcal{B}=({\mathcal{B}}_{1},\dots ,{\mathcal{B}}_{k})$ is an ordered basis for $V$ and the matrix of $f$ with respect to $\mathcal{B}$ is $$\left(\begin{array}{ccc}{A}_{1}& \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & {A}_{k}\end{array}\right)$$ where ${A}_{i}$ is the matrix of ${f}_{{W}_{i}}$ with respect to ${\mathcal{B}}_{i}$. |

Let $b={t}^{3}-10{t}^{2}+23t-14$ and $a={t}^{4}-3{t}^{3}+3{t}^{2}-3t+2$. Find $d=\mathrm{gcd}(a,b)$ and find $x,y\in \mathbb{Q}\left[t\right]$ such that $d=ax+by$. | |

Let $b={t}^{3}-6{t}^{2}+t+4$ and $a={t}^{5}-6t+1$. Find $d=\mathrm{gcd}(a,b)$ and find $x,y\in \mathbb{Q}\left[t\right]$ such that $d=ax+by$. | |

The eigenvalues of the (linear transformation corresponding to the) matrix $$A=\left(\begin{array}{ccc}2& 1& 3\\ 0& -1& 4\\ 0& 0& 0\end{array}\right)$$ satisfy $\mathrm{det}(A-\lambda I)=0$. Determine the eigenvalues and show that the coresponding eigenspaces are dimension 1 and are generated by the eigenvectors $$\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\phantom{\rule{2em}{0ex}}\left(\begin{array}{c}1\\ -3\\ 0\end{array}\right),\phantom{\rule{2em}{0ex}}\left(\begin{array}{c}-7\\ 8\\ 2\end{array}\right)$$ | |

Let ${C}^{\infty}\left(\mathbb{R}\right)$ be the space of functions $f\mathbb{R}\to \mathbb{R}$ which are differentiable infinitely often. Show that the eigenvectors of differentiation are the functions ${e}^{ax}$, for $a\in \mathbb{R}$. Determine the eigenvalues. | |

Suppose that a linear transformation on ${\mathbb{R}}^{3}$ has matrix $$A=\left(\begin{array}{ccc}3& 2& 0\\ 1& 1& 0\\ 0& 0& 3\end{array}\right)$$ with respect to the basis $\{{e}_{1}=(1,0,0),{e}_{2}=(0,1,0),{e}_{3}=(0,0,1\left)\right\}$. Show that the subspace $W=\mathrm{span}\{{e}_{1},{e}_{2}\}$ is $f$-invariant and that the matrix of ${f}_{W}$ with respect to the basis $\{{e}_{1},{e}_{2}\}$ is $$\left(\begin{array}{cc}3& 2\\ 1& 1\end{array}\right).$$ | |

Show that, in ${\mathbb{R}}^{3}$, a complement to a plane through the origin is any line through the origin which does not lie in the plane. | |

Show that, in ${\mathbb{R}}^{4}$, the subspaces $\mathrm{span}\left\{\right(1,0,0,0),(0,1,0,0\left)\right\}$ and $\mathrm{span}\left\{\right(0,0,1,0),(0,0,0,1\left)\right\}$ are complementary. | |

Show that, in $\mathbb{R}\left[x\right]$, the subspaces $\mathrm{span}\{2,1+x,1+x+{x}^{3}\}$ and $\mathrm{span}\{{x}^{2}+3{x}^{4},{x}^{4},{x}^{5},{x}^{6},\dots \}$ are complementary. | |

Let $f$ be a linear transformation with matrix $\left(\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 0\end{array}\right).$ Show that the minimal polynomial of $f$ is $(x-2)(x-3)x$. | |

Find the minimal polynomial of the matrix $\left(\begin{array}{cc}2& 0\\ 3& -1\end{array}\right).$ | |

Find the minimal polynomial of the matrix $\left(\begin{array}{cc}0& 1\\ 1& -1\end{array}\right).$ | |

Find the minimal polynomial of the matrix $\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right).$ | |

Find the minimal polynomial of the matrix $\left(\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 0& 0& 1\end{array}\right).$ | |

Show that the matrices $$\left(\begin{array}{cccc}2& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 1\end{array}\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left(\begin{array}{cccc}2& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$ have the same minimal polynomial and different characteristic polynomial. | |

Show that the matrix $$A=\left(\begin{array}{ccc}1& -3& 3\\ 3& -5& 3\\ 6& -6& 4\end{array}\right)$$ has minimal polynomial ${x}^{2}-2x-8$. Use this to determine the inverse of $A$. | |

Show that a linear transformation $f$ is invertible if and only if its minimal polynomial has non-zero constant term. Assuming $f$ is invertible, how can the inverse be calculated if the minimal polynomial is known? | |

Suppose that $A$ is an $n\times n$ upper triangular matrix with zeroes on the diagonal. Prove that ${A}^{n}=0$. | |

Let $f$ be a linear transformation on a vector space $V$ with minimal polynomial ${x}^{2}-1$. Suppose that $2\ne 0$ in the field of scalars. (Thus, for example, $\mathbb{Z}/2\mathbb{Z}$ is not allowed as the field of scalars.) Show directly that the subspaces $$\{v\in V\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}f\left(v\right)=v\left\}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\right\{v\in V\phantom{\rule{0.5em}{0ex}}\left|\phantom{\rule{0.5em}{0ex}}f\right(v)=-v\}$$ are complementary subspaces of $V$. Find a diagonal matrix representing $f$. | |

Let ${\mathcal{P}}_{n}\left(\mathbb{R}\right)$ be the vector space of polynomials in $\mathbb{R}\left[x\right]$ of degree $\le n$. Show that the linear transformation ${\mathcal{P}}_{n}\left(\mathbb{R}\right)\to {\mathcal{P}}_{n}\left(\mathbb{R}\right)$ given by differentiation with respect to $x$ cannot be represented by a diagonal matrix. | |

Let $T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{3}$ be the linear transformation defined by $T(x,y)=(x+2y,-x,0)$. Find the matrix of $T$ with respect to the (ordered) bases $B=\left\{\right(1,3),(-2,4\left)\right\}$ for ${\mathbb{R}}^{2}$ and $C=\left\{\right(1,1,1),(2,2,0),(4,0,0\left)\right\}$ for ${\mathbb{R}}^{3}$. | |

Let $V$ be the subspace of functions from $\mathbb{R}$ to $\mathbb{R}$ spanned by $\{{e}^{2t},t{e}^{2t},{t}^{2}{e}^{2t}\}$. Show that differentation with respect to $t$ is well defined linear transformation $D$ on $V$ and find the matrix of $D$ with respect to the basis $\{{e}^{2t},t{e}^{2t},{t}^{2}{e}^{2t}\}$ of $V$. | |

Find the minimal polynomial of the matrix $\left(\begin{array}{cc}-3& 2\\ -2& 1\end{array}\right).$ | |

Find the minimal polynomial of the matrix $\left(\begin{array}{ccc}2& 1& 1\\ 0& 3& 2\\ 0& 0& 2\end{array}\right).$ | |

Find the minimal polynomial of the matrix $\left(\begin{array}{ccc}2& 1& 2\\ 0& 3& 2\\ 0& 0& 2\end{array}\right).$ | |

Let ${W}_{1}$ and ${W}_{2}$ be subspaces of a vector space $V$. Show that $V$ is the direct sum of ${W}_{1}$ and ${W}_{2}$ if and only if every vector $v\in V$ can be written uniquely in the form $v={w}_{1}+{w}_{2}$, where ${w}_{1}\in {W}_{1}$ and ${w}_{2}\in {W}_{2}$. | |

- (i) Show that the complex numbers $\u2102$ is a vector space over the field of real numbers $\mathbb{R}$.
- (ii) Show that $\{1,i\}$ is a basis for $\u2102$ over $\mathbb{R}$.
- (iii) Let $\alpha =a+ib$ be a complex number. Show that multiplication by $\alpha $ is a linear transformation $f:\u2102\to \u2102$. Find the matrix of $f$ with respect to the basis $\{1,i\}$.
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Let $f:V\to V$
be a linear transformation on an $n$-dimensional
vector space with minimal polynomial $m\left(x\right)={x}^{n}$.
- (i) Show that there is a vector $v\in V$ such that ${f}^{n-1}\left(v\right)\ne 0$.
- (ii) Show that $B=\{$ {f}^{n-1}\left(v\right),$ {f}^{n-2}\left(v\right),\dots $ {f}^{2}\left(v\right),$ f\left(v\right),v\}$is\; a\; basis\; for$ V$.$$$$
- (iii) Find the matrix for $f$ with respect to the basis $B$.
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Find a linear transformation $f:V\to V$ on an infinite dimensional vector space $V$ which satisfies no monic polynomial equation $p\left(f\right)=0$. |

[Ar]
M. Artin, *Algebra*, Prentice-Hall, 1991.

[GH]
J.R.J. Groves and
C.D. Hodgson,
*Notes for 620-297: Group Theory and Linear Algebra*, 2009.

[Ra]
A. Ram,
*Notes in abstract algebra*, University of Wisconsin, Madison 1994.