MA251 - Algebra I

Assignment 2 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 2 NOVEMBER (Monday of the fifth 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. Show that a $2\times 2$-matrix $A$ with $A^3 = 0$ also satisfies $A^2 = 0$.

P2. Let $A$ be an $8\times 8$-matrix $A$ over $\mathbb{R}$, and suppose that $c_A(z) = (1-z)^8$ and $\mu_A(z) = (z-1)^4$ . Write down the possible JNFs for $A$. How would you decide which was the correct JNF?

P3. Is it true that for all $n\times n$-matrices over complex numbers $A$ and $B$ the JNFs of $AB$ and $BA$ are the same? Give a proof or counterexample.

P4. Let $q:V \rightarrow K$ be a quadratic form. Prove that, for all $\ensuremath{\mathbf{u}}\xspace ,\ensuremath{\mathbf{v}}\xspace ,\ensuremath{\mathbf{w}}\xspace \in V$,

$$q(\ensuremath{\mathbf{u}}\xspace +\ensuremath{\mathbf{v}}\xsp... ...nsuremath{\mathbf{w}}\xspace )=\ensuremath{\mathbf{0}}\xspace .$$

P5. Write down the symmetric matrices corresponding to the quadratic forms

$$\hbox{(i) } 3x^2-7xy+11y^2;\hspace{1cm} \hbox{(ii) } xy+yz+xz;\hspace{1cm} \hbox{(iii) } w^2-xy+z^2.$$

P6. A bilinear form $\tau:V \times V \rightarrow K$ is called alternating or anti-symmetric
if $\tau(\ensuremath{\mathbf{u}}\xspace ,\ensuremath{\mathbf{v}}\xspace ) = -\tau(\ensuremath{\mathbf{v}}\xspace ,\ensuremath{\mathbf{u}}\xspace )$ for all $\ensuremath{\mathbf{u}}\xspace ,\ensuremath{\mathbf{v}}\xspace \in V$.

(i) Show that the form $\tau$ is alternating if and only if $\tau(\ensuremath{\mathbf{v}}\xspace ,\ensuremath{\mathbf{v}}\xspace )=0$ for all $\ensuremath{\mathbf{v}}\xspace \in V$;

(ii) Show that any bilinear form on $V$ is equal to the sum of a symmetric form and an alternating form.

P7. An $n\times n$ matrix is called orthogonal if $A^\mathrm TA = I_n$. Let $A$ be an orthogonal matrix over $\mathbb{R}$.

(i) Prove $\det(A) = \pm 1$.

(ii) Prove that, if $\lambda$ is an eigenvalue of $A$ with $\lambda \in \mathbb{R}$, then $\lambda = \pm 1$.
(Hint: Transpose the equation $A\ensuremath{\mathbf{v}}\xspace = \lambda \ensuremath{\mathbf{v}}\xspace$.)

P8. Let $A=\left( \begin{array}{rr} 2&1\ 1&2\end{array} \right)$, with real coefficients.

(i) Using either JNF or Lagrange's interpolation, compute $A^n$ explicitly for a natural number $n$.

(ii) Find a polynomial $f(Z)$ of degree less than $2$ such that $e^{tA}=f(A)$ and compute $e^{tA}$ explicitly.

(iii) Solve the system of differential equations

$$\left\{ \begin{array}{lcrrr} x^\prime (t)&=&2x(t)&+&y(t)\\ y^\prime (t)&=&x(t)&+&2y(t) \end{array}\right. ,$$

with initial condition $x(0)=1$ and $y(0)=-1$

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 $A=\left( \begin{array}{rr} 3&-2\\ 2&-2\end{array} \right)$, with real coefficients.

(i) Using either JNF or Lagrange's interpolation, compute $A^n$ explicitly for a natural number $n$. [2 marks]

(ii) Find a polynomial $f(Z)$ of degree less than $2$ such that $e^{tA}=f(A)$ and compute $e^{tA}$ explicitly. [2 marks]

(iii) Solve the system of differential equations

$$\left\{ \begin{array}{lcrrr} x^\prime (t)&=&3x(t)&-&2y(t)\\ y^\prime (t)&=&2x(t)&-&2y(t) \end{array}\right. ,$$

with initial condition $x(0)=2$ and $y(0)=1$ [1 mark]

2. Let $D:\mathbb{R}[X] \rightarrow \mathbb{R}[X]$ be the differentiation operator $D (f(X)) = f^\prime (X)$. Prove that $e^{tD}(f(X))=f(X+t)$ for a real number $t\in \mathbb{R}$. [3 marks]

3. (i) Write down the symmetric matrix $A$ corresponding to the quadratic form $q(\ensuremath{\mathbf{v}}\xspace ) = wz-xy$ in the 4 variables $w,x,y,z$. [1 mark]

(ii) Find a change of coordinates to transform $q$ to the form $\alpha w_1^2 + \beta x_1^2 + \gamma y_1^2 + \delta z_1^2$. [2 marks]

(iii) Write down the corresponding change of basis matrix $P$, and verify that $P^\mathrm TAP$ is diagonal. [2 marks]

4. Let $\tau:W \times V \rightarrow \mathbb{F}$ be a bilinear map, where $V$ and $W$ are vector spaces over a field $\mathbb{F}$. Recall that the dual space is denoted by $V^\ast$. Let $U$, $U_i$ be subspaces of $V$.

(i) Prove that $U^\perp$, defined by

$$U^\perp = \{ \ensuremath{\mathbf{w}}\xspace \in W \mid \tau(\... ...0}}\xspace \forall \ensuremath{\mathbf{u}}\xspace \in U \},$$

is a subspace of $W$. [1 marks]

(ii) Prove that $U \subseteq (U^\perp)^\perp$ [1 marks]

(iii) Prove that $U_1 \subseteq U_2$ implies $U_2^\perp \subseteq U_1^\perp$ and [1 marks]

(iv) Show that the map $T_U:W \rightarrow U^\ast$ defined by $T_U(\ensuremath{\mathbf{w}}\xspace )(\ensuremath{\mathbf{u}}\xspace ) = \tau(\ensuremath{\mathbf{w}}\xspace ,\ensuremath{\mathbf{u}}\xspace )$ for $\ensuremath{\mathbf{w}}\xspace \in W, \ensuremath{\mathbf{u}}\xspace \in U$ is a linear map from $W$ to $U^\ast$, and that $\ker(T_U) = U^\perp$. [2 marks]

(v) Deduce that $\dim(U) + \dim(U^\perp) \ge \dim(W)$. [2 marks]