# Coordinate mapping of diagonal $M_{2x2}$ matrices

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 $$P$$.

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