Reposted: Linear Transformation and Dimensions

Hi, I am having troubles even starting this problem and any help would be appreciated.

**1)** The question is: If B is an nxn matrix, X = {A in M(mxn) matrix space | AB = 0} and Y = {AB | A in M(mxn) matrix space}, show that X and Y are subpaces of M(mxn) and that dim(X) + dim(Y) = mn.

**What I have)** From observation, I can see that X resembles a null space or kernel of a T, and Y resembles an image space or image of a T. And partially, I can see that dim(X) + dim(Y) = mn = dim(M(mxn) matrix space).

**2) **As a related question, can someone explain to me how to **show** if a transformation is one to one and/or onto? Suppose T: V-->W: I know that one to one means that the ker(T) = {0}, does that mean when a T is one to one, there are no vectors that meet the requirement T(v) = 0? And likewise, for onto, not every vector w in W can be mapped via T(v)?

As always, thank you for any help. My main focus is the initial problem, and the explanation for one to one/onto is secondary; if there are any sources you may point me to for the explanation, please do - I just hope for a different way to approach the thinking since I am quite confused.

As for the reason for repost: The thread got buried after I posted late last night and am desperate to get some help with this.

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Re: Reposted: Linear Transformation and Dimensions

Re: Reposted: Linear Transformation and Dimensions

1) X=ker(T) and Y=Im(T), where T(A)=AB. I think this will help you do the rest.

2) It means the only solution to T(v)=0 is v=0. If you want to test if T is one to one you solve the equation T(v)=0 and see if there are any v different than zero. if there are, then it is not one to one. If there aren't, T is one to one.

Onto means that every vector can w can be expressed as T(v), for some v in V. You can also show that T is onto by showing that dim Im(T) = dim W.

Re: Reposted: Linear Transformation and Dimensions

Thank you very much hedi and ModusPonens, you have relieved much of my stress.

If you would not mind checking this as well, to see if I used one-to-one/onto correctly:

Suppose U -> V -> W, have linear transformations, from U to V is T, from V to W is S.

Show that if ST is one-to-one, and T is onto, then S is one-to-one:

(ST)(u) = 0 where u = 0, T(u) = v

S(T(u)) = S(v) = 0

S(v) = 0

I'm just not sure if I have proved or said anything.

Re: Reposted: Linear Transformation and Dimensions

T is onto so for v there is u such that Tu=v.so,Sv=0 implies STu=0 .Now use one-to-one of S to conclude that u, hence also v ,is 0.

Re: Reposted: Linear Transformation and Dimensions

Just a correction of a typo in hedi's post:

*T is onto so for v there is u such that Tu=v.so,Sv=0 implies STu=0 .Now use one-to-one of S***T** to conclude that u, hence also v ,is 0.