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Let $D$ be the set of diagonal $M_{2x2}$ matrices

$1)$ Show that $D$ is a subspace of $M2×2$. To prove $D$ is a subspace it must follow that: $$D \in \overrightarrow{0}$$ $$\overrightarrow{v_1} + \overrightarrow{v_2} \in D$$ $$C\overrightarrow{v_1} \in D$$ Let $D =\begin{bmatrix} a & 0 \\\ 0 & b \end{bmatrix}$ where $a,b=0$ then $D$ becomes $\begin{bmatrix} 0 & 0 \\\ 0 & 0 \end{bmatrix}$

Let $\overrightarrow{v_1} = \begin{bmatrix} a_1 & 0 \\\ 0 & b_1 \end{bmatrix}$ and $\overrightarrow{v_2} = \begin{bmatrix} a_2 & 0 \\\ 0 & b_2 \end{bmatrix}$ where $a_1,b_1,a_1,b_2 \in \mathbb{R}$ $$\overrightarrow{v_1} + \overrightarrow{v_2} = \begin{bmatrix} a_1 + a_2 & 0 \\\ 0 & b_1+ b_2 \end{bmatrix} \therefore \overrightarrow{v_1} + \overrightarrow{v_2} \in D$$

Let $C \in \mathbb{R}$ $$C\overrightarrow{v_1}= \begin{bmatrix} Ca_1 & 0 \\\ 0 & Cb_1 \end{bmatrix} \therefore C\overrightarrow{v_1} \in D $$

$2)$ What is the dimension of $D$?

Looking at the basis of $D$ we get: $$Span =\left(\begin{bmatrix}1& 0 \\\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0&0 \\\ 0& 1 \end{bmatrix}\right)$$ which is a 2 element linearly independent set, therefore the dimension of $D=2$.

$3)$ Consider a coordinate mapping $T: D \to \mathbb{R}^{dim(D)}$. Explain what $T\left(\begin{bmatrix} a & 0 \\\ 0 & b \end{bmatrix} \right)$ is.

I said that $T$ is the transformation that maps $D$ to $\mathbb{R}^2$ which is the basis $\left( \begin{bmatrix} 1 & 0 \\\ 0 & 1 \end{bmatrix} \right)$ of $\mathbb{R}^2$. This doesn't feel right to me because in the next part I'm asked to find this exact matrix.

$4)$ Find the ”standard basis”, $\varepsilon$, for $D$ and then find another basis, $\beta$, for $D$.

The standard basis $\varepsilon$ is just the elementary matrix $\begin{bmatrix} 1 & 0 \\\ 0 & 1 \end{bmatrix}$. I'm just guessing here but $\beta$ is any scalar multiple of $\varepsilon$ because $\varepsilon$ is already a basis and therefore contains as many vectors as needed to span the set of $D$.

$5)$ Find the change of coordinate matrix $P_{\beta}$.

I'm not sure if $P$ is supposed to be the set of all polynomials with degree $\le 1$ and haven't chosen a basis $\beta$ yet because I wanna make sure I'm correct in my assumptions but when I have $\beta$ the change of coordinates will something look like: $$C_1*\overrightarrow{b_1}+C_2*\overrightarrow{b_2}= \begin{bmatrix} 1 \\\ x \end{bmatrix}$$

where $C_1$ and $C_2$ are the coordinates relative to $P$.

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  • $\begingroup$ For 3: $T(D)=(a,b)$. For 3-5: a basis must have two elements, as you computed the dimension (and actually a basis). $\endgroup$
    – Anthony
    Oct 30, 2023 at 20:41

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