Suman Cole

2021-01-28

Let W be the subspace of all diagonal matrices in ${M}_{2,2}$ . Find a bais for W. Then give the dimension of W.

If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix

If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix

Tasneem Almond

Skilled2021-01-29Added 91 answers

Step 1

Given that W is the subspace of all diagonal matrices in${M}_{2,2}$

The objective is to a basis for W.

Step 2

Consider the given vector space W of all diagonal matrices in${M}_{2,2}$

Therefore,

$W=\{\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right],\text{a and b can be any real number}\}$

Let E be basis for W.

Therefore,

$\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]=a\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+b\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$

Now, let${\lambda}_{1}$ and ${\lambda}_{2}$ be any two scalars such that:

${\lambda}_{1}\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+{\lambda}_{2}\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]=[]$

$\left[\begin{array}{cc}{\lambda}_{1}& 0\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& {\lambda}_{2}\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\left[\begin{array}{cc}{\lambda}_{1}& 0\\ 0& {\lambda}_{2}\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

Equating the elements:

${\lambda}_{1}=0,{\lambda}_{2}=0$

Therefore,$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$ and $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$ are linear independent.

Hence, the basis of W is$E=\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\}$ and dimension is 2.

Given that W is the subspace of all diagonal matrices in

The objective is to a basis for W.

Step 2

Consider the given vector space W of all diagonal matrices in

Therefore,

Let E be basis for W.

Therefore,

Now, let

Equating the elements:

Therefore,

Hence, the basis of W is

Jeffrey Jordon

Expert2022-01-30Added 2605 answers

Answer is given below (on video)

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