# Harrison Chapman

## Linear Algebra I: Homework 5

Due: Friday, September 22, 2017
1. Find the matrix for the linear transformation $\tfrac{d}{dx}$ acting on the vector space $V = \{a+bx+cx^2 \mid a,b,c \in \mathbb R\}$ of polynomials of degree $2$ or less in the ordered basis $B = (x^2, x, 1)$.

We plug the basis vectors in $B$ into $\tfrac d{dx}$ and evaluate using calculus. We then write them as column vectors with regards to $B$. We then stack them side-by-side into a matrix.

So the matrix is

2. Use your matrix from (a) to rewrite the differential equation ($p(x)$ is a vector in $V$)

as a matrix equation. Find all solutions of the matrix equation, then write them as elements of $V$.

Using our answer to (a), the equation becomes the matrix equation with unknown vector $p(x)$;

Using Gaussian elimination we get that

As there is no pivot in the third column of the matrix, the coefficient on $1$ is free. The pivots in columns two and one respectively say that the coefficient on $x$ is $0$ and the coefficient on $x^2$ is $1/2$. So,

Where $c$ can be any real number.

This is the same as the antiderivative $\int {x \, dx}$, as you might have expected.

3. Find the matrix for the linear transformation $\tfrac{d}{dx}$ acting on the same vector space $V$ but now with the ordered basis $C = (x^2+x,x^2-x,1)$.

We plug the basis vectors in $C$ into $\tfrac d{dx}$ and evaluate using calculus. We then write them as column vectors with regards to $C$. We then stack them side-by-side into a matrix.

So the matrix is

4. Use your matrix from (c) to rewrite the differential equation ($p(x)$ is a vector in $V$)

as a matrix equation. Find all solutions of the matrix equation, then write them as elements of $V$.

Using our answer to (b), the equation becomes the matrix equation with unknown vector $p(x)$;

Using Gaussian elimination we get that

As there is no pivot in the third column of the matrix, the coefficient on $1$ is free. The pivots in columns two and one respectively say that the coefficient on $x^2-x$ is $1/4$ and the coefficient on $x^2+x$ is $1/4$. So,

Where $c$ can be any real number.

Notice how this answer simplifies to $p(x) = \tfrac 12 x^2 + c$ just like we got for the answer to (b). We’ll see this later, how we get the same answer no matter the basis that we use.

1. Suppose that $A$ is a square matrix that is antisymmetric, meaning that

Prove that $\mathop{\rm tr}(A) = 0$.

Properties of trace have that

But since $A = -A^T$ this means that

which can only happen if $\mathop{\rm tr}(A) = 0$.

2. The matrix exponential of a matrix $M$ is given by the Taylor series,

For $\lambda \in \mathbb R$, let $A$ be the matrix:

and let $B$ be the matrix:

1. Find a concise formula for $A^k$, for any integer $k$.

Since $A = \lambda I$,

2. Find a concise formula for $\exp(A)$.

3. Find $B^k$, for any integer $k$.

$B$ has no inverse, so $B^{k}$ is undefined for any negative $k$.

$B^0$ is defined to be $I$, and $B^1 = B$.

$B^2 = 0$ by an easy calculation, so for any $k \ge 2$, $B^k = B^2B^{k-2} = 0B^{k-2} = 0$.

4. Find a concise formula for $\exp(B)$.

Most stuff is zero, so the answer is:

3. Find the determinant of the matrix

4. Is the determinant of a matrix a linear transformation into the real numbers $\mathbb R$? Explain or give a counterexample.

No. There are many reasons, but one example is for the $2 \times 2$ identity matrix $I$. We know that $\det(I) = 1$ but $\det(2I) = 4$ rather than $2\det(I) = 2$. So $\det$ cannot be a linear transformation.