MA251 - Algebra I

Assignment 1 October 2009

Answer the questions on your own paper. Write your own name in the top left-hand corner, and your university ID number in the top right-hand corner. Use the problems at the beginning as well as exercises in the lecture notes for a warm up. Solutions to the FOUR TEST problems must be handed in by 15.00 on MONDAY 19 OCTOBER (Monday of the third week of term), or they will not be marked. There will be an award of 5 extra marks for clarity, so do a good job.

These are practice problems for you to sharpen your teeth on.

P1. Let $V=\mathbb{R}_{\leq 2}[x]$ be the vector space of polynomials of degree at most 3 over $\mathbb{R}$. Let $\ensuremath{\mathbf{e}}\xspace _1,\ensuremath{\mathbf{e}}\xspace _2,\ensuremath{\mathbf{e}}\xspace _3$ and $\ensuremath{\mathbf{e}}\xspace _1^\prime,\ensuremath{\mathbf{e}}\xspace _2^\prime,\ensuremath{\mathbf{e}}\xspace _3^\prime$ be the bases $1,x,x^2$ and $1+x^2, \; 1+2x, \; (1-x)^2$ of $V$.

(i) Write down the matrices of linear maps $\ensuremath{\mathbf{e}}\xspace _i \rightarrow \ensuremath{\mathbf{e}}\xspace _i^\prime$ and $\ensuremath{\mathbf{e}}\xspace _i^\prime \rightarrow \ensuremath{\mathbf{e}}\xspace _i$.

(ii) Using the matrices found in (i), write the polynomial $3-x+x^2$ as a linear combination of $\ensuremath{\mathbf{e}}\xspace _1^\prime,\ensuremath{\mathbf{e}}\xspace _2^\prime,\ensuremath{\mathbf{e}}\xspace _3^\prime$.

(iii) Write the linear map $T:V\rightarrow V$ given by $T(f(x)) = f^{\prime \prime} (x)$ in the basis $\ensuremath{\mathbf{e}}\xspace _1,\ensuremath{\mathbf{e}}\xspace _2,\ensuremath{\mathbf{e}}\xspace _3$.

(iv) Using the matrices found in (i), write $T$ in the basis $\ensuremath{\mathbf{e}}\xspace _1^\prime,\ensuremath{\mathbf{e}}\xspace _2^\prime,\ensuremath{\mathbf{e}}\xspace _3^\prime$.

P2. Find the eigenvalues of each of the following matrices $A$ over the complex numbers $\mathbb{C}$. For each eigenvalue find one corresponding eigenvector, and then write down a matrix $P$ such that $P^{-1}AP$ is diagonal.

(i) $\left( \begin{array}{rr} 1&-2 1&4\end{array} \right)$, (ii) $\left( \begin{array}{rr} 1&-1 2&3\end{array} \right)$, (iii) $\left( \begin{array}{rr} \cos \theta&-\sin \theta\\ \sin \theta&\cos \theta\end{array} \right)$, (iv) $\left( \begin{array}{rr} \cos \theta&\sin \theta\\ \sin \theta&-\cos \theta\end{array} \right)$.

P3. Find the eigenvalues of the following pairs of matrices, and use them to decide which of the pairs are similar.

(i) $\left( \begin{array}{rr} 3&2 1&7\end{array} \right)$ and $\left( \begin{array}{rr} 1&2 3&4\end{array} \right)$;

(ii) $\left( \begin{array}{rrr}8&3&-6 2&1&0 10&4&-7\end{array} \right)$ and $\left( \begin{array}{rrr}1&0&0 -2&2&0 4&10&-1\end{array} \right)$;

(iii) (harder) $\left( \begin{array}{rrr}2&1&0 0&2&1 0&0&-1\end{array} \right)$ and $\left( \begin{array}{rrr}2&0&0 0&2&0 0&-1&-1\end{array} \right)$.

P4. Say whether the following statements are true or false. If true, briefly state why. If false, give a counterexample. In all parts, $A=(\alpha_{i,j})$ is an $n\times n$ matrix over a field $\mathbb{K}$ containing $\frac{1}{2}$, the trace of $A$ is the sum of diagonal elements: $\mbox{tr} (A) = \sum_{i=1}^n \alpha_{i,i}$. .
(i) If $A$ is skew-symmetric, that is $A^T=-A$, then $\mbox{tr}(A)=0$.
(ii) If $n$ is odd and $A$ is skew-symmetric then $\det (A)=0$.
(iii) If $A^3 = I_n$ then $\det (A)=1$.
(iv) If $A^2=A$ then $A$ is singular.
(v) If $n=2$ then $A^2-\mbox{tr}(A)A + \det (A) I_2 = A^2-\mbox{tr}(A)A -\frac{1}{2} \mbox{tr} (A^2) I_2+\frac{1}{2} \mbox{tr} (A)^2 I_2 = 0$.

P5. Compute the following determinants. You may want to use elementary row and/or column operations to reduce the matrix to a simpler form first.

(i) $\left\vert \begin{array}{rrrr}1&0&1&-1 1&1&0&0\\ 1&0&1&1 1&0&0&1\end{array} \right\vert$; (ii) $\left\vert \begin{array}{rrrr}1&2&0&0 2&1&3&0\\ 1&-1&0&2 0&-2&1&1\end{array} \right\vert$; (iii) $\left\vert \begin{array}{rrrrr} 1&2&1&-2&-1 2&0&1&0&1 -3&0&8&0&-1 3&0&0&-2&1 -3&-5&-3&1&2\end{array} \right\vert$;
(iii) $\left\vert \begin{array}{rr} \cos (\theta)&-\sin(\theta) \ \sin(\theta)&\cos(\theta)\end{array} \right\vert$ where $\theta$ is some real number.

P6. Write down a matrix in JCF with minimal polynomial $(x + 1)(x - 2)^3 (x + 3)^2$ and characteristic polynomial $(1 + x)^2 (2 - x)^4 (3 + x)^3$ .

P7. Let $T:V\rightarrow V$ be a linear map, where $\mbox{dim}(V ) = n$, and suppose that $T^n = 0$ and that there exists a vector $v \in V$ with $T^{n-1}(v) \neq 0_V$ . Prove that the vectors $v,\; T (v), \; T^2 (v), \ldots \; T^{n-1} (v)$ are linearly independent and that the nullity of $T$ is $1$.

P8. For the following matrices $A$, find minimal polynomials, matrices $B$, and invertible matrices $P$, such that $P^{-1}AP = B$ is in Jordan Canonical Form.

$$(i) \left(\!\!\begin{array}{rr} 4&1 -1&6 \end{array}\!\!\r... ...} 0&1&1 2&1&-1 -6&-5&-3 \end{array}\!\!\right)\hspace{1cm}$$

$$(iii)\ \left(\!\!\begin{array}{rrr} 10&-4&-8 8&-2&-8 8&... ...&1&0&0 0&0&1&0 0&0&0&1 -1&0&-2&0 \end{array}\!\!\right)$$

The following problems are test problems for you to submit for marking. Write concise but complete solutions only to the questions asked. Additional 5 marks are awarded for clarity.

1. Let $\alpha_1, \alpha_2, \ldots, \alpha_n \in \mathbb{K}$, where $n \geq 2$, $\mathbb{K}$ is a field. You will have to prove that

$$\left\vert \begin{array}{rrrrr} 1&\alpha_1&\alpha_1^2&\ldots&... ...} \right\vert = \prod_{1 \leq i < j \leq n}(\alpha_j-\alpha_i).$$

(i) Show it for $n=2$. [1 mark]

(ii) Show it for $n=3$. [1 mark]

(iii) Use induction to show it for general $n$. (Hint: Use elementary column operations to reduce the first row to $(1,0,0\ldots)$, then expand by the first row.) [3 marks]

2. For the following matrices $A$, find minimal polynomials, matrices $B$, and invertible matrices $P$, such that $P^{-1}AP = B$ is in Jordan Canonical Form.

$$(i)\ \left(\!\!\begin{array}{rr} 15&-4\ 49&-13 \end{array}\... ...&1&0\ 1&-2&1&0\ 1&-1&0&0 \end{array}\!\!\right)\hspace{1cm}$$

[2 marks each]

3. Let $T:V\rightarrow V$ be a linear map for which $T^k = 0$ (the zero map) for some $k > 0$. (Linear maps with this property are called nilpotent.) Prove that 0 is an eigenvalue of $T$, and that it is the only eigenvalue of $T$. [4 marks]

4. Let $V$ be a vector space over a field $\mathbb{K}$ with a basis $\ensuremath{\mathbf{e}}\xspace _i$, $i\in I$. The dual vector space $V^\ast$ is defined as a set of all linear maps $V\rightarrow \mathbb{K}$. While elements of $V$ are called vectors, elements of $V^\ast$ should be called covectors. Given the basis of $V$ as above, we define a covector $\ensuremath{\mathbf{e}}\xspace ^i\in V^\ast$, $i\in I$ by $\ensuremath{\mathbf{e}}\xspace ^i (\ensuremath{\mathbf{e}}\xspace _j) = \delta_{ij}$ (1 if $i=j$, 0 otherwise).

(i) Prove that the covectors $\ensuremath{\mathbf{e}}\xspace ^i$, $i\in I$ are linear independent. [1 mark]

(ii) Consider $T\in V^\ast$ defined by $T (\ensuremath{\mathbf{e}}\xspace _i) = 1$ for all $i$. Assuming that $V$ is finite-dimensional, prove that the covectors $\ensuremath{\mathbf{e}}\xspace ^i$ form a basis and write $T$ explicitly as a linear combination of the covectors $\ensuremath{\mathbf{e}}\xspace ^i$. [1 mark]

(iii) Assuming that $V$ is infinite-dimensional, prove that $T$ is not a linear combination of $\ensuremath{\mathbf{e}}\xspace ^i$. (It implies that they do not form a basis). [1 mark]

(iv) Assume that $V$ is finite-dimensional. From what we prove it follows that both $V$ and $V^\ast$ are vector spaces of the same dimension. Consider a linear bijection $\phi : V \rightarrow V^\ast$ defined by $\phi (\ensuremath{\mathbf{e}}\xspace _i)=\ensuremath{\mathbf{e}}\xspace ^i$. Show that this bijection depends on the original choice of basis. (Hint: Consider 2 different bases in a one dimensional vector space and compute bijections explicitly ) [2 marks]