# Harrison Chapman

## Linear Algebra I: Homework 12

Due: Monday, December 4, 2017

Notice: This final homework assigment is due after our final midterm, but the questions on it cover material that will be on the test. I encourage you to work these problems before the test, although you don’t need to have a set of solutions written up until after.

1. Let $P_n$ be the vector space of polynomials in $x$ of degree at most $n$, and let $\frac d{dx}$ be the derivative (a linear transformation on $P_n$).

1. What is the rank of $\frac d{dx}$?

If you’ve already answered part (b), the answer is $\mathop{\rm rank} \frac{d}{dx} = \dim{\mathop{\rm domain} \frac d{dx}} - \mathop{\rm null} \frac{d}{dx} = \dim{P_n} - 1 = (n+1) - 1 = n$.

If you haven’t: Derivatives of polynomials of degree at most $n$ have to be polynomials of degree at most $n-1$. So, $\mathop{\rm rank}\frac{d}{dx} = \dim{P_{n-1}} = n$.

1. What is the nullity of $\frac d{dx}$?

If you’ve already answered part (a), the answer is $\mathop{\rm null} \frac{d}{dx} = \dim{\mathop{\rm domain} \frac d{dx}} - \mathop{\rm rank} \frac{d}{dx} = \dim{P_n} - n = (n+1) - n = 1$.

If you haven’t: The only polynomials which differentiate to 0 are constant polynomials (and all constant polynomials differentiate to 0). So, $\mathop{\rm null}\frac d{dx} = \dim{P_0} = 1$.

2. Let $P_3$ be the vector space of polynomials in $x$ of degree at most    $3$. Then $\langle f,g \rangle = \int_{0}^{10}{4f(x)g(x)\;dx}$ is an    inner product on $P_3$. If $B$ is an orthonormal basis for $P_3$ under this inner product, compute:

There’s not enough information to solve this question using the formula for the inner product (what on earth is $B$?). However, since $B$ is orthonormal, the formula for the inner product of two $B$-vectors is just the usual dot product formula, $6 - 0 - 3 + 8 = 11$.

3. Find an orthonormal basis for the kernel of the matrix $M$:

To find a basis for the kernel, we solve $M\vec x = \vec 0$ by Gaussian elimination to find,

It turns out that $B$ is already orthogonal (why?) so we just have to normalize it:

4. For the following parts, a matrix or linear transformation is described. Explain whether or not it is invertible.

1. The linear transformation $K$ diagonalizes as

It is not, since 0 is an eigenvalue of $K$ (why?) based on its diagonalization.

2. The linear transformation $L: \mathbb R^n \to \mathbb R^n$ scales volumes of hypercubes by a factor of $1/2$.

The determinant of a transformation is the quantity which expresses how it scales the volumes of hypercubes. So, $\det L = 1/2$, which is not zero, so $L$ is invertible.

3. The rank of the linear transformation $H: \mathbb R^7 \to \mathbb R^7$ is 5.

For a matrix to be invertible, it must be full rank. However, $5 \le 7$, so $H$ is not full rank, and so not invertible.

4. The linear transformation $Q: \mathbb R^m \to \mathbb R^m$ is surjective.

A surjective, square, linear transformation is invertible. So $Q$ is.

5. Find the component of the velocity vector,

in the direction as the vector,

The component of a vector $\vec v$ in the direction of $\vec w$ is the scaling multiple of $\vec w$ in the orthogonal decomposition of $\vec v$:

6. Find the least squares solution to the equation,

The least squares solution to an equation $M\vec x = \vec b$ is the solution to $M^TM\vec x = M^T\vec b$. We have that $M^TM$ is,

and $M^T\vec b$ is,

Solving the equation

yields the least squares solution,