Question

TRUE OR FALSE

There exist a basis of ${\mathbf{ℝ}}^{\mathbf{2}\mathbf{\times }\mathbf{2}}$that consists of orthogonal matrices.

Short answer

The given statement is true.

Step-by-step solution

Definition of Orthogonal matrix

Matrix A is said to be orthogonal of and only if $A{A}^T=I$.

Explanation of the solution

Consider the statement as follows:

“If there exists a basis of $R^{2\times 2}$that consists of orthogonal matrices.

Consider the matrices as follows:

role="math" localid="1659625186673" $\left\{A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right],C=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],D=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\right\}$

Clearly, the above matrices are orthogonal.

Since orthogonal vectors are linearly independent and the matrices are linearly independent.

Therefore, $\left\{A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right],C=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],D=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\right\}$is the basis of $R^{2\times 2}$

Hence, the given statement “If there exists a basis of $R^{2\times 2}$that consists of orthogonal matrices is true.