Chardonnay Felix

2021-01-31

Find a basis for the space of $2×2$ diagonal matrices.

Jayden-James Duffy

Step 1
let the space of $2×2$ diagonal matrices be as
$\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$
we can choose values of a and b randomly Step 2
Basis of matrix $\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$ can be taken as
a=1,b=0
$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$
a=0, b=1 $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$

Don Sumner

We need to identify a collection of linearly independent matrices that cover the space of 2*2 diagonal matrices in order to construct a basis for the space.
$D=\left[\begin{array}{cc}{d}_{1}& 0\\ 0& {d}_{2}\end{array}\right]$
where ${d}_{1}$ and ${d}_{2}$ are arbitrary constants.
To form a basis, we need to find distinct diagonal matrices that are linearly independent. Here's a possible basis for the space of 2×2 diagonal matrices:
$B=\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}$
We have two distinct diagonal matrices, and we can see that no linear combination of one matrix can produce the other. Therefore, these matrices are linearly independent.
To confirm that these matrices span the space, we can take an arbitrary 2×2 diagonal matrix:
$D=\left[\begin{array}{cc}{d}_{1}& 0\\ 0& {d}_{2}\end{array}\right]$
We can rewrite this matrix as a linear combination of the basis matrices:
$D={d}_{1}\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+{d}_{2}\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$
This shows that any 2×2 diagonal matrix can be expressed as a linear combination of the basis matrices. Therefore, the set $B$ is a basis for the space of 2×2 diagonal matrices.

karton

To find a basis for the space of 2$×$2 diagonal matrices, we can start by considering the general form of a diagonal matrix:
$\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$
where $a$ and $b$ are scalars.
We can see that any 2$×$2 diagonal matrix can be written as a linear combination of the following matrices:
$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$
Let's call these matrices ${D}_{1}$, ${D}_{2}$, and ${D}_{3}$, respectively. We can express any 2$×$2 diagonal matrix as:
$a{D}_{1}+b{D}_{2}+c{D}_{3}$
where $a$, $b$, and $c$ are scalars.
Therefore, the set $\left\{{D}_{1},{D}_{2},{D}_{3}\right\}$ forms a basis for the space of 2$×$2 diagonal matrices.
Alternatively, we can express the basis as:
$\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\right\}$

user_27qwe

A set of matrices that span the space and are linearly independent must be found in order to establish a basis for the space of 2x2 diagonal matrices.
A 2x2 diagonal matrix can be represented as:
$\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$
where $a$ and $b$ are any real numbers.
To find a basis, we can consider two diagonal matrices with different values for $a$ and $b$. A possible basis for the space of 2x2 diagonal matrices can be:
${D}_{1}=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\phantom{\rule{0ex}{0ex}}{D}_{2}=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right].$
These two matrices span the space of 2x2 diagonal matrices since any diagonal matrix can be written as a linear combination of ${D}_{1}$ and ${D}_{2}$.
To show that the basis is linearly independent, we need to verify that the only solution to the equation ${c}_{1}{D}_{1}+{c}_{2}{D}_{2}=\mathbf{0}$ (where $\mathbf{0}$ represents the zero matrix) is ${c}_{1}={c}_{2}=0$.
$\left[\begin{array}{cc}{c}_{1}& 0\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& {c}_{2}\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
This implies ${c}_{1}={c}_{2}=0$. Hence, the basis $B=\left\{{D}_{1},{D}_{2}\right\}$ is linearly independent.
Therefore, the basis for the space of 2x2 diagonal matrices is given by:
$B=\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}.$

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