# Linear Algebra Dot Product - Subspace/Vector Space

enter image description here I'm trying to solve parts b and c for this textbook practice problem and I can't seem to understand how to construct the basis for either of them. Would part b be a subspace because they are multiplying by vector v to get the zero vector?

• Part b is indeed a subspace, because it is the null space of the linear map $v \mapsto v\cdot a$. This should also give you a hint regarding its dimension. – Bungo Oct 18 '17 at 17:11
• Therefore the zero vector has dimension zero. Does that mean that the basis would just be an empty set for part B? – CluelessCoder Oct 18 '17 at 17:21
• The dimension of the null space of the linear map $v \mapsto v \cdot a$ is $3$, not $0$. This is because the dimension of the image is $1$ (as a subspace of $\mathbb R$ it could only be $0$ or $1$; it's not $0$ because, for example, $a\cdot a \neq 0$). Then rank-nullity tells us that the null space must have dimension $4 - 1 = 3$. So, your basis should have three vectors. – Bungo Oct 18 '17 at 17:24