For Fixed Matrices $R, S$, the Matrices $RAS$ form a Subspace

Vector Space Problems and Solutions

Problem 664

Let $V$ be the vector space of $k \times k$ matrices. Then for fixed matrices $R, S \in V$, define the subset $W = \{ R A S \mid A \in V \}$.

Prove that $W$ is a vector subspace of $V$.

 
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Proof.

We verify the subspace criteria: the zero vector of $V$ is in $W$, and $W$ is closed under addition and scalar multiplication.


First, let $\mathbf{0} \in V$ be the $k\times k$ zero matrix. Then $ R \mathbf{0} S = \mathbf{0}$, and so $\mathbf{0} \in W$.


Now suppose $X, Y \in W$. Then there are elements $A, B \in V$ such that $RAS = X$ and $RBS = Y$. Then
\[X + Y = RAS + RBS = R (A+B) S\] and so $X+Y \in W$.


Now for a scalar $c \in \mathbb{R}$ and matrix $X = RAS \in W$, we have
\[cX = c RAS = R (cA )S,\] and so $cX \in W$ as well.


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