# Harrison Chapman

## Linear Algebra I: Homework 7

Due: Friday, October 13, 2017
1. Consider the set of vectors in $\mathbb R^3$:

1. Find a vector in $S$ which can be expressed as a linear combination of the other vectors in $S$.

Any vector is a fine answer here. For example,

2. Make a new set of vectors $T$ by removing your vector from part (a) from $S$. Is $T$ linearly independent? Explain.

Based on our answer to (a), we’d get

This set isn’t linearly dependent, since,

3. Find a vector in $T$ which can be expressed as a linear combination of the other vectors in $T$.

From our discussion in (b)

4. Make a new set of vectors $U$ by removing your vector from part (c) from $T$. Is $U$ linearly independent? Explain.

Based on our answer to (c), we’d get

which is a linearly independent set of vectors since the matrix,

has reduced row echelon form

that has a pivot in every column.

2. A unit vector is a vector whose magnitude is 1.

1. Describe all unit vectors $\vec x$ in $\mathbb R^2$.

There are a few different ways to express this answer. One way is any vector of the form,

for any $\theta$.

2. For which unit vectors $\vec x$ is

a basis for $\mathbb R^2$?

Any answer to part (a) except

works, since the determinant of the matrix

is $\sin \theta$. That is, it is a basis so long as $\theta \ne \pi n$.

1. Find a basis for the vector space of diagonal $2 \times 2$ matrices.

There are many correct answers. One standard one is,

2. An upper triangular matrix is a matrix whose entries below diagonal entries are all 0. Find a basis for the vector space of upper triangular $2 \times 2$ matrices.

There are many correct answers. One standard one is,

3. Consider the two bases for $\mathbb R^2$;

1. Find a matrix $M$ that changes column vectors for basis $B$ into column vectors for basis $C$.

The matrix,

changes $B$-vectors into standard vectors, and the matrix

changes $C$-vectors into standard vectors. On the other hand, $M_C^{-1}$ changes standard vectors into $C$-vectors.

So the matrix $M_C^{-1} M_B$ changes $B$-vectors into $C$-vectors.

2. Find a matrix $N$ that changes column vectors for basis $C$ into column vectors for basis $B$.

We want to do the opposite of what we did in (a). This means, we want the inverse of our answer: $(M_C^{-1} M_B)^{-1} = M_B^{-1}M_C$ is the matrix that changes $C$-vectors into $B$-vectors.

4. Is 5 an eigenvalue of the matrix:

Yes, it is. This is because, if we plug in $5$ into the characteristic polynomial of the matrix, we get the determinant of the matrix,