Let's find its dimension

Algebra Level 4

If $V$ is a vector space of dimension 100, and $W_1$ and $W_2$ are vector sub spaces of $V$ of dimension 59 and 63 respectively, then the maximum possible dimension of $W_1+ W_2$ is:

The answer is 100.

1 solution

Otto Bretscher
Feb 3, 2016

Let $v_1,..,v_{100}$ be a basis of $V$ . If $W_1=span(v_1,..,v_{59})$ and $W_2=span(v_{38},..v_{100})$ , then $W_1+W_2=V$ so $\dim(W_1+W_2)=\dim(V)=\boxed{100}$ .

i showed that dim(W1+W2)<= minimum{ dimV, dimW1+dimW2} So minimum =100

Amar Mavi - 5 years, 4 months ago

but how did you show that?

Otto Bretscher - 5 years, 4 months ago

As dim[w1+w2] = dim[w1]+dim[w2]-dim[w1 /\ w2] , so dim[w1]<= dim[w1+w2]<= dim[w1]+dim[w2] and dim[w2]<=dim[w1+w2]<=dim[V] then we imply that max{dim[w1],dim[w2]}<=dim[w1+w2]<=min{dim[V],dim[w1]+dim[w2]}....

Amar Mavi - 5 years, 4 months ago

@Amar Mavi – I don't get it. It seems to me that $\dim(W_1+W_2)\leq \min\left(\dim(V),\dim(W_1)+\dim(W_2)\right)$ merely implies that $\dim(W_1+W_2)\leq 100$ . You need to show that 100 is attained; that's what I'm trying to do in my solution.

Otto Bretscher - 5 years, 4 months ago

@Amar Mavi – Is there a way you can write this up in LaTeX?

Plus I love the picture of Cayley, there is a bust of him in my unis mathematics department.

Isaac Buckley - 5 years, 4 months ago

Let's find its dimension

The answer is 100.

1 solution

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