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Found in: Page 176

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Find the basis of all 2X2 diagonal matrix, and determine its dimension.

The dimension of a 2X2 diagonal matrix is 2 which is spanned by $Span\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}.$.

See the step by step solution

## Step 1: Determine the matrix.

Consider the matrix $A=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$ where and are real.

Simplify the equation $A=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$as follows.

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

## Step 2: Find the basis of each required space

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

Therefore, the matrix A is spanned by $Span\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}$.

Hence, the dimension of A is 2 and spanned by $Span\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}$.