# 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 $$W_1 \subseteq W_2$$?

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## Answers and Replies

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.