Master Linear Algebra Proofs: Dimension Theorem, Rank-Nullity Theorem, and More!

[killpoppop]

Hey people. I find myself getting through my course but currently with not as much understanding as I would like. We've got to some proofs and i either vaguely understand them or do not know how to prove them.

Homework Statement

The first would be to prove the Dimension theorem that.

dimU + dimV = dim(U + V) + dim( U intersection V )

The second is if U,V are finite dimensional vector spaces over a field, and T: V -> W is a bijection

Show dimU = dimW

The last is to do with the Rank-Nullity Theorem and a basic linear transformation T: Rn -> Rm

That if m < n prove that T is not injective
And if m = n prove that T is injective iff T is surjective.

Some general insight into these proofs would be very very helpful. All feedback would be very much appreciated.

Thanks.

 

[kurnimaha]

The first would be to prove the Dimension theorem that.

dimU + dimV = dim(U + V) + dim( U intersection V )

Let {u₁, ... , u_k} be a basis for U and {v₁, ... , v_p} a basis for V. Now consider the intersection U$$\cap$$V. If U$$\cap$$V = {0}, what can you say about the linear depence of the set {u₁, ... , u_k, v₁, ... , v_p}? How about the situation U$$\cap$$V $$\neq$$ {0}?

The second is if U,V are finite dimensional vector spaces over a field, and T: V -> W is a bijection

Show dimU = dimW

If T: V -> W is a bijective map, then what can you say about its range and kernel? How is this related to the first problem?

The last is to do with the Rank-Nullity Theorem and a basic linear transformation T: Rn -> Rm

That if m < n prove that T is not injective
And if m = n prove that T is injective iff T is surjective.

Again, consider range and kernel.

 

[HallsofIvy]

Hey people. I find myself getting through my course but currently with not as much understanding as I would like. We've got to some proofs and i either vaguely understand them or do not know how to prove them.

Homework Statement

The first would be to prove the Dimension theorem that.

dimU + dimV = dim(U + V) + dim( U intersection V )

I would do this: subtract dim(U intersect V) from both sides to get the equivalent
dim(U+ V)= dimU + dimV- dim( U intersection V )
Choose a basis for U intersect V, then extend it to a basis of U. Extend that same basis for U intersect V to a basis for V. Show that the union of those two bases is a basis for U+ V.

The second is if U,V are finite dimensional vector spaces over a field, and T: V -> W is a bijection

Show dimU = dimW

Choose a basis for U, {u1, u2, ..., un} and show that {Tu1, Tu2, ... Tun} are independent. Thus, $dimU\le dimW$. Choose a basis for W, {w1, w2, ..., wm} and show that {T^-1w1, T^-1w2, ..., T^-1wn} are independent.

The last is to do with the Rank-Nullity Theorem and a basic linear transformation T: Rn -> Rm

That if m < n prove that T is not injective
And if m = n prove that T is injective iff T is surjective.

Again choose a basis for Rn, {v1, v2, ..., vn} and look at {Tv1, Tv2, ..., Tvn}.

Some general insight into these proofs would be very very helpful. All feedback would be very much appreciated.

Thanks.

 

[killpoppop]

Thanks for the feedback it all helps. But still doesn't clear a lot up for me.

Choose a basis for U intersect V, then extend it to a basis of U. Extend that same basis for U intersect V to a basis for V. Show that the union of those two bases is a basis for U+ V.

For this questions I've proved U+V is a subspace. What your saying is the next step but it doesn't make much sense to me. Can you explain further?

 

[killpoppop]

Also can anyone suggest a decent book or website where i can read up on these?