Linear Algebra dimensions proof

[zcd]

Homework Statement

Let W₁ and W₂ be subspaces of a finite-dimensional vector space V. Determine necessary and sufficient conditions on W₁ and W₂ so that $$dim(W_1 \cap W_2)=dim(W_1)$$

Homework Equations

Replacement Theorem

The Attempt at a Solution

To clarify on the question: is the problem asking for conditions such that $$condition\iff dim(W_1 \cap W_2)=dim(W_1)$$?
If it is, is it possible to say that let dim(W₁)=m and dim(W₂)=n, n≥m and W₁ is nested inside W₂, so $$W_1 \subseteq W_2$$?

 

[ystael]

To clarify on the question: is the problem asking for conditions such that $$condition\iff dim(W_1 \cap W_2)=dim(W_1)$$?

Yes.

If it is, is it possible to say that let dim(W₁)=m and dim(W₂)=n, n≥m and W₁ is nested inside W₂, so $$W_1 \subseteq W_2$$?

"$$W_1 \subseteq W_2$$" is a reasonable guess for what the right condition might be, but what you have written is not a proof of either direction of the implication.