Linear Algebra

Is the Set of All Orthogonal Matrices a Vector Space?

Problem 611

An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$.
Let $V$ be the vector space of all real $2\times 2$ matrices.

Consider the subset
\[W:=\{A\in V \mid \text{$A$ is an orthogonal matrix}\}.\] Prove or disprove that $W$ is a subspace of $V$.

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Solution.

We claim that $W$ is not a subspace of $V$.

One way to see that $W$ is not a subspace of $V$ is to note that the zero vector $O$ in $V$, which is the $n\times n$ zero matrix, is not in $W$ as we have $O^{\trans}O=O\neq I$.
Thus, $W$ is not a subspace of $V$.

Another approach

You may also show that scalar multiplication (or addition) is not closed in $W$.

For example, the identity matrix $I$ is orthogonal as $I^{\trans}I=I$, and thus $I$ is an element in $W$.
However, the scalar product $2I$ is not orthogonal since
\[(2I)^{\trans}(2I)=4I\neq I.\]

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