Bases and transformations

Given the following bases vectors for $\mathbb{R}^2$:

$$ \begin{array}{l} 1. \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\} \\[12pt] 2. \left\{ \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \begin{bmatrix} -1 \\ 2 \end{bmatrix} \right\} \\[12pt] 3. \left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \end{bmatrix} \right\} \\ \end{array} $$

1. Which of these sets forms a basis for $\mathbb{R}^2 $? Justify your answer by checking linear independence and spanning the vector space.

Consider three bases in $\mathbb{R}^2$:

1. The standard basis $$ E = \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\} $$
2. The basis $$ B = \left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ -1 \end{bmatrix} \right\} $$
3. The basis $$ C = \left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \end{bmatrix} \right\} $$

Given two vectors in the standard basis:

$$ \text{v} = \begin{bmatrix} 3 \\ 1 \end{bmatrix}, \text{w} = \begin{bmatrix} 7 \\ 6 \end{bmatrix} $$

1. Draw the basis vectors for $E$, $B$ , and $C$ and the vectors $\text{v}$ and $\text{w}$ on a piece of paper.

2. Express $\text{v}$ and $\text{w}$ in the basis $B$.

3. Express $\text{v}$ and $\text{w}$ in the basis $C$.

4. Find the change of basis matrix $T_E^B$ that given a vector expressed in the basis E maps it to the basis B.

5. Find the change of basis matrix $T_E^C$ that given a vector expressed in the basis E maps it to the basis C.

6. Use the change of basis matrices to transform $\text{v}$ and $\text{w}$ from standard basis to the basis $B$ and $C$ respectively. Verify that the resulting coordinates match your results from the previous steps.

Let $ T: \mathbb{R}^2 \to \mathbb{R}^2 $ be a linear transformation represented by the matrix: $$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$ in the standard basis. Let $$ B = \left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 2 \end{bmatrix} \right\} $$ be a new basis.

1. Calculate the matrix that represents $ T $ with respect to the basis $ B $. Use the change of basis matrices to show your work.

Suppose $ \mathbb{R}^3 $ has a basis given by $$ B = \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} \right\}. $$

1. Verify that $ B $ is a basis by showing it spans $ \mathbb{R}^3 $ and is linearly independent.
2. If a fourth vector $ \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix} $ is added to $ B $, does it still form a basis for $ \mathbb{R}^3 $? Explain your reasoning in terms of dimension.