Linear algebra and column spaces

[Niles]

Homework Statement

I have a linear transformation T which is associated with a matrix A. I want to find the image of T, which is given by the column space of A. I have done this.

Now I have a vector B = (x,y,z,w)^T, and I want the vector B to be part of the image of T. To do this, I write the matrix P, which is given by (A|B). Do I just set this matrix equal zero and solve it to find the image of T, wherein the matrix B is?

 

[HallsofIvy]

That sounds to me like you would be solving the equation AB= 0- which would give the kernel of T, not its image.

I'm not clear on what your question is. you say you have already found the image of T, now you want to find a vector B in the image of T? Any one of the columns of A corresponds to a vector that is in the image of T.

 

[Niles]

I want to find the image of T, where the columnvector B is part of the image of T.

What I do is that I write (A|B), bring to rref and find the columnspace from that. What I wrote in my original post was a mistake. But from what you wrote ("Any one of the columns of A corresponds to a vector that is in the image of T"), I guess my method will work?

 

[Dick]

I want to find the image of T, where the columnvector B is part of the image of T.

What I do is that I write (A|B), bring to rref and find the columnspace from that. What I wrote in my original post was a mistake. But from what you wrote ("Any one of the columns of A corresponds to a vector that is in the image of T"), I guess my method will work?

It will work if you realize what you are doing. You are adding B to the image by defining a new linear transformation T' corresponding to the matrix (A|B) with domain in a space one dimension higher than T. You can't 'add' anything to the range of T itself. Is that what you want to do?

 

[Niles]

Yes, that is exactly what I want to do. Your explanation cleared it for me - thanks, and congratulations with your award.

 

[Dick]

Thanks! And you're welcome!