Answer:
No, H is not a subspace of the vector space V.
Step-by-step explanation:
A matrix is a rectangular array in which elements are arranged in rows and columns.
A matrix in which number of columns is equal to number of rows is known as a square matrix.
Let H denote set of all 2×2 idempotent matrices.
H is a subspace of a vector space V if [tex]u+v \in H[/tex] for [tex]u,v \in V[/tex] and [tex]cu \in H[/tex].
Let [tex]A=\begin {pmatrix}1&0\\0&1 \end{pmatrix}[/tex]
As [tex]A^2=A\times A=\begin {pmatrix}1&0\\0&1 \end{pmatrix}\begin {pmatrix}1&0\\0&1 \end{pmatrix}=\begin {pmatrix}1&0\\0&1 \end{pmatrix}=A[/tex], A is idempotent.
So, [tex]A \in H[/tex]
[tex]A+A=\begin {pmatrix}1&0\\0&1 \end{pmatrix}+\begin {pmatrix}1&0\\0&1 \end{pmatrix}=\begin {pmatrix}2&0\\0&2\end{pmatrix} \\ \left ( A+A \right )^2=\begin {pmatrix}2&0\\0&2\end{pmatrix}\begin {pmatrix}2&0\\0&2\end{pmatrix}=\begin {pmatrix}4&0\\0&4\end{pmatrix}\neq A[/tex]So, A+A is not idempotent and hence, does not belong to H.
So, H is not a subspace of the vector space V.
