NLA MT25, Subspaces
Flashcards
In what way does a matrix $V \in \mathbb R^{n \times d}$ represent a subspace $\mathcal S$?
The span of the columns.
@Prove that if:
- $V _ 1 \in \mathbb R^{n \times d _ 1}$
- $V _ 2 \in \mathbb R^{n \times d _ 2}$
- $V _ 1, V _ 2$ both have linearly independent column vectors
- $d _ 1 + d _ 2 > n$
then:
- There is a nonzero intersection between the two subspaces $\mathcal S _ 1 = \text{span}(V _ 1)$ and $\mathcal S _ 2 = \text{span}(V _ 2)$.
Consider the matrix $M := [V _ 1, V _ 2]$, which is of size $n \times (d _ 1 + d _ 2)$. Since $d _ 1 + d _ 2 > n$, this matrix has a right null vector $c \ne 0$ such that $Mc = 0$. Then splitting $c = \begin{bmatrix}c _ 1 \ -c _ 2\end{bmatrix}$ gives the required result, since $V _ 1 c _ 1 = V _ 2 c _ 2$.