Question

Let A be an n×n matrix. Then Show that the set, U={uϵRn : Au = - 3un} is a Subspace ofRn

Answer 1

Answer on Question #43815 – Math – Linear Algebra

Let $A$ be an $n \times n$ matrix. Then Show that the set, $U = \{u \in \mathbb{R}^n : Au = -3un\}$ is a Subspace of $\mathbb{R}^n$.

Solution.

Let $V$ be a vector space over field $K$. Suppose that $W$ is a subset of $V$. If $W$ is a vector space itself (which means that it is closed under operations of addition and scalar multiplication), with the same vector space operations as $V$ has, then $W$ is a subspace of $V$. Then $W$ is a subspace of $V$ if and only if $W$ satisfies the following condition:

If $x \in W$ and $y \in W$ then $ax + by \in W$ for all $a, b \in K$.

In our case $V = R^n$, $W = U$ and $K = R$. Let's verify the previous condition

Let $x \in U$ and $y \in U$ then $Ax = -3nx$ and $Ay = -3ny$.

Let's consider the combination $u = ax + by$

$$Au = A(ax + by) = aAx + bAy = a(-3nx) + b(-3ny) = -3(ax + by)n = -3un, \text{ so we get}$$

$Au = -3un$, hence $U$ is a subspace of $R^n$.

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