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
for
and
.
Let ![A=\begin {pmatrix}1&0\\0&1 \end{pmatrix}](https://tex.z-dn.net/?f=A%3D%5Cbegin%20%7Bpmatrix%7D1%260%5C%5C0%261%20%5Cend%7Bpmatrix%7D)
As
, A is idempotent.
So, ![A \in H](https://tex.z-dn.net/?f=A%20%5Cin%20H)
So, A+A is not idempotent and hence, does not belong to H.
So, H is not a subspace of the vector space V.