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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 $Span\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}$.