I this question from an old exam:
Determine all vectors located in Cola) and Col (B). Explain why all the vectors contained in both Col (A) and Col (B) form a subspace iR3 and calculate its dimension.
A below:
B below:
The thing is i understand the calculations but what i don't understand is that they take
the cross-product of the vectors in Matrix A and same thing in Matrix B.
And after that they take Cross-product of the both vectors and get a another 90 degrees
vector and say this vector is both Col(A) and Col(B) how come that the Cross-Product can give a vectors in both spaces i don't understand i am very confused normally you use the Cross-Product to calculate a Equation for a plane?
Solution below:
Why can cross-product solve this problem i don't understand i am very confused ??
Best Answer
Two 2-dimensional subspaces of $\mathbb{R}^3$ are either identical or have a 1-dimensional intersection. Clearly these two subspaces aren't identical, so we just need to find one vector in there intersection to find them all.
$v\times w, v, w$ forms a basis of $\mathbb{R}^3$ whenever $v, w$ are linearly independent. Furthermore, any vector orthogonal to $v\times w$ lies in the span of $v, w.$
So, $(v\times w) \times (v'\times w')$ will lie in the intersection of the subspace spanned by $v, w$ and the subspace spanned by $v', w'.$ Since the cross product is not 0 in this example, it must be spanning since the intersection in your case is 1-dimensional.