Question

Prove that if A and B are subspaces of Rⁿ, then AnB is also a subspace of Rⁿ

Answer 1

Given that,

A is subspace of $R^n$

B is subspace of $R^n$

But we know that $A\cap B$ means element is common in both set A and B.

As per the intersection theory $A\cap B=\{x| x \in A \& x\in B \}$

The zero vector $0 \ of \ R^n \ \in A \cap B$

For all the sum $x,y\in A \cap B,$$x+y \in A\cap B$

For all and $x∈A\cap B$ and $r \in R$ , we have $rx\in A∩B$

.As here A and B are the subspace of $R^n$ , the zero vector 0 is in both A and B

Hence the $0\in R^n$ is also lies in $A\cap B$

.