Question

In V3(R), Find dim(A+B) and dim(A^B) where A is the subspace spanned by (1,1,1) and B is the subspace spanned by (-1,-1,-1).

Answer 1

Question 1.

In $V_{3}(\mathbb{R})$ find $\dim(A+B)$ and $\dim(A\cap B)$ where $A$ is the subspace spanned by $(1,1,1)$ and $B$ is the subspace spanned by $(-1,-1,-1)$.

Solution.

Note that the vectors $(1,1,1)$ and $(-1,-1,-1)$ are linearly dependent, namely, $(-1,-1,-1)=-(1,1,1)$. Therefore, the $1$-dimensional subspaces spanned by these vectors coincide. More precisely, each vector of $A$ has the form $\alpha(1,1,1)$ for some $\alpha\in\mathbb{R}$ and this vector also belongs to $B$, because $\alpha(1,1,1)=-\alpha(-1,-1,-1)$. And conversely, each vector of $B$ lies in $A$ by the similar reason. Thus, $A=B$ and so

$A+B$ $=A+A=A,$

$A\cap B$ $=A\cap A=A.$

Hence, $\dim(A+B)=\dim(A\cap B)=\dim A=1$.

Answer: $\dim(A+B)=\dim(A\cap B)=1$. ∎