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.