Assignment #9

Quiz 9 in discussion section Tuesday, November 27:

For a linear transformation $T:V\rightarrow W$ between vectors spaces and a subspace $S$ of $W$, (a) be able to define what $T^{-1}(S)$ means, and (b) be able to prove that $T^{-1}(S)$ is a subspace of $V$. (See Problem O-3 and its solution.)

Assignment due Friday, November 30.

where	Do but don't hand in	Hand in
U	7
CC	CC-1, CC-6, CC-7, CC-8, CC-9	CC-2, CC-3, CC-4, CC-5, CC-10
DD	DD-1, DD-3, DD-5, DD-7	DD-2, DD-4, DD-6, DD-8, DD-9

Problem CC-1. For the linear transformation $L _ A:$R$^ 2 \rightarrow$   R$^ 2$ with $A = \left[\begin{array}{rr}3&-1\\ -3&1\end{array}\right]$, sketch (a) the range, (b) the null space, and (c) the inverse image of $\left[\begin{array}{r}2\\ -2\end{array}\right]$ [the set of vectors in the domain that go to this vector]. Use two sets of axes. Do a careful job, indicating the scale by ``tick marks'' on the axes. Label your answers.

Problem CC-2. Let $V$ be the solution space of the DE $y'' = -4y$. Let $B$ be the basis consisting of $\cos 2x, \sin 2x$. If $T:V\rightarrow V$ is the differentiation operator, find the matrix of $T$ relative to this basis.

Problem CC-3. (a) Diagonalize the matrix $A = \left[\begin{array}{rr}2&1\\ 2&3\end{array}\right]$. Be sure to give $P$ and $D$, but you don't need to find $P ^ {-1}$.

(b) Diagonalize $A ^ 2$. (Again, give $P$.)

Problem CC-4. (a) Diagonalize $A = \left[\begin{array}{rr}2&3\\ 6&5\end{array}\right]$, showing the matrices involved, with explicit entries, including finding a matrix inverse where needed. Show that your answer checks, by multiplying out.

(b) Find a matrix $B$ with $B ^ 3 = A$.

(Method: Diagonalize $A$ using an appropriate $P$, take a cube root, and ``undiagonalize'' by doing a similarity back using $P ^ {-1}$ in place of $P$.)

Problem CC-5. Show that for an $m \times n$ matrix $A$ and $n \times m$ matrix $B$ over the same field, trace$(AB) =$   trace$(BA)$.

Problem CC-6. (a) Show that square matrices $A,B$ of the same size, if at least one of $A$ and $B$ is invertible (i.e., nonsingular) then $AB$ and $BA$ are similar. (Method: Easy matrix manipulation.)

(b) Looking through past homework, find an example to show that (a) might not hold if neither of $A$ and $B$ is invertible. (Method: What if $AB = 0$?)

Problem CC-7. Invent a specific numeric example of each of the following, giving a brief reason in each case:

(a) A $2 \times 2$ matrix with eigenvalues whose sum is 1 and whose product is $-1$.

(b) A $2 \times 2$ diagonal matrix that is a rotation but is not the identity matrix.

(c) A matrix $A$ such that $A ^ 2 - A + I = 0$. (Method: Just find a $2 \times 2$ matrix $A$ with characteristic polynomial $\lambda ^ 2 - \lambda + 1$. Magically, any matrix acts like a root of its own characteristic polynomial. Check your answer.)

(d) A matrix $A$ and eigenvalue $\lambda$ so that the eigenspace $E _ \lambda$ has dimension 2.

Problem CC-8. By ``permutation'' we will always mean a permutation on a finite number of symbols. Recall that a transposition means a 2-cycle such as $\left(\begin{array}{rr}1&3\end{array}\right)$.

(a) Show that the n-cycle $\left(\begin{array}{rrrr}1&2,\dots, n\end{array}\right)$ is a product of transpositions. In other words, you can achieve an $n$-cycle by switching two symbols at a time, several times. (Advice: Try this for $n = 3$, then $n = 4$, etc. until you see a general method.)

(b) Show that every permutation is a product of transpositions. (Quote the fact that every permutation is a product of ``disjoint'' cycles and use (a).)

(c) A permutation $\sigma$ is said to be even if $\sigma$ is the product of an even number of transpositions, or odd if $\sigma$ is the product of an odd number of transpositions.

It is a fact that any permutation is either even or odd, but not both.

Looking at the multiplication table for $S_3$ (p. AA 3), say which permutations are even, which are odd, and check that a transposition times an even permutation is odd and vice versa. (It may help to draw a horizontal line and vertical line to split the table into four areas.)

Problem CC-9. (a) For functions $x(t), y(t)$, solve the system of differential equations (DE's)

$$\left \{ \begin{array}{rrrr} x' & = & 4 x & \\ y' & = & & -7 y \\ \end{array}\right .$$ with $x(0) = 1$, $y(0) = 2$.

(Method: Recall that for a single dependent variable the solution of $y' = k y$ is $y = y(0) e^{kt}$.)

(b) Rewrite the problem in matrix form. (Method: x$' = D$   x with x$(0) = ??$. What is $D$?)

(c) Show that the solution can be expressed in matrix form using a matrix power, as in Problem W-2, except that the initial values become a vector on the right: x$(t) = ??$   x$(0)$.

(d) Check the solution directly by taking the derivative of a matrix power that is a function of $t$, inventing any reasonable rules you need for derivatives of matrix functions of $t$.

Problem CC-10. Solve the system of DE's x$' = A$x where $A = \left[\begin{array}{rr}3&2\\ 2&3\end{array}\right]$ and x$(0) = \left[\begin{array}{r}1\\ 2\end{array}\right]$, two ways:

(a) by using similarity;

(b) by using a matrix power.

(Methods: For (a) diagonalize $A$ by $P^{-1} A P = D$ and substitute x$= P$z, where z$= \left[\begin{array}{r}z\\ w\end{array}\right]$. Use algebra to get the system of DE's z$' = D$z and also find z$(0)$. Solve this diagonal system and then put things back in terms of x. The solution does not have to be in matrix form.

For (b) just propose a solution and then check it by taking the derivative, inventing any needed rules as in the problem CC-.)