The vectors a1, a2, a3, b1, b2, b3 are given by a1=, a2=, a3=, b1=, b2=, b3=. 1) The subspace of R4 spanned by a1, a2, a3 is denoted by V1, and the subspace of R4 spanned by b1, b2, b3 is denoted by V2. Show that V1 and V2 each have dimension 3. 2) The set of vectors which belong to both V1 and V2 is denoted by V3. Find a basis for V3.

Question

The vectors a1, a2, a3, b1, b2, b3 are given by a1=<3,2,1,0>, a2=<1,1,0,0>, a3=<0,0,1,0>, b1=<3,2,0,2>, b2=<2,2,0,1>, b3=<1,1,0,1>. 1) The subspace of R4 spanned by a1, a2, a3 is denoted by V1, and the subspace of R4 spanned by b1, b2, b3 is denoted by V2. Show that V1 and V2 each have dimension 3. 2) The set of vectors which belong to both V1 and V2 is denoted by V3. Find a basis for V3.

anonymous · Answer

I have figured out the first question #1.  By making $$V1=\left[\begin{matrix}3 & 1 & 0 \ 2 & 1 & 0 \ 1 & 0 & 1\ 0 & 0 & 0\end{matrix}\right]$$$$V2=\left[\begin{matrix}3 & 2 & 1 \ 2 & 2 & 1 \ 0 & 0 & 0\ 2 & 1 & 1\end{matrix}\right]$$ and putting both into reduced row echelon form, found that they both have a dimension of 3. However, I'm confused how to proceed for question 2.

anonymous · Answer

I have also attached a copy of the original question from a CIE Further Maths Pure Exam.