Linear Algebra dimensions proof

In summary, the problem is asking for necessary and sufficient conditions on W1 and W2 such that the dimension of their intersection is equal to the dimension of W1. The Replacement Theorem may be used to clarify the question. One possible condition is that W1 is a subspace of W2, written as W_1 \subseteq W_2. However, this is not a complete proof and further exploration may be needed.
  • #1
zcd
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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 [tex]dim(W_1 \cap W_2)=dim(W_1)[/tex]

Homework Equations


Replacement Theorem

The Attempt at a Solution


To clarify on the question: is the problem asking for conditions such that [tex]condition\iff dim(W_1 \cap W_2)=dim(W_1)[/tex]?
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 [tex]W_1 \subseteq W_2[/tex]?
 
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  • #2
zcd said:
To clarify on the question: is the problem asking for conditions such that [tex]condition\iff dim(W_1 \cap W_2)=dim(W_1)[/tex]?

Yes.

zcd said:
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 [tex]W_1 \subseteq W_2[/tex]?
"[tex]W_1 \subseteq W_2[/tex]" 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.
 

1. What is the definition of dimensions in linear algebra?

The dimensions in linear algebra refer to the number of independent vectors that span a vector space. It represents the minimum number of vectors required to express any vector in that space.

2. How do you prove that a set of vectors is linearly independent?

A set of vectors is considered linearly independent if none of the vectors in the set can be expressed as a linear combination of the other vectors. This can be proven by setting up a system of equations and solving for the coefficients of the linear combination. If the only solution is when all coefficients are equal to zero, then the vectors are linearly independent.

3. What is the difference between basis and dimension?

Basis refers to a set of linearly independent vectors that span a vector space, while dimension refers to the number of vectors in that basis. In other words, the dimension of a vector space is the number of vectors in its basis.

4. Can a vector space have more than one basis?

Yes, a vector space can have multiple bases. However, all bases for a given vector space will have the same number of vectors, which is the dimension of that vector space.

5. How is the dimension of a vector space related to its span?

The dimension of a vector space is equal to the minimum number of vectors required to span that space. In other words, the dimension is the number of vectors in the smallest basis for that vector space.

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