Solve Column Space, Matrix Problem with (x,y,z,w)^T

[Niles]

[SOLVED] Column space, matrix

Homework Statement

I have a linear transformation f from R^4 -> R^4 given by a matrix. I have to find the range of f(R^4) which containts the vector (x,y,z,w)^T.

The Attempt at a Solution

I know that the range of f is the column space, how do I make sure that the vector (x,y,z,w)^T is part of the range?

Thanks in advance.

 

[HallsofIvy]

I have no idea what you mean by "the vector (x,y,z,w)^T". That appears to be just a general vector, perhaps telling you that you can use x, y, z, and w in your formulas describing the range. Or are you given specific values for those variables?

Yes, you can get the range by looking at the span of the columns. For example, if your matrix is
$$\left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 0 \end{array}\right]$$
then the range is the three dimensional subspace spanned by (1, 0, 0, 0)^T, (0, 1, 0, 0)^T, and (0, 0, 1, 0)^T, which can be written simply "w= 0".

 

[Niles]

I am not given a specific vector, but for this example we can equal it to (1,2,3,1) (just an example!). The text says "find the range of f(R^4) that has the above vector included".
The way you did it, you expressed the range in terms of that vector. I am quite sure that it is not what they are asking for. Are we supposed to write it in the form of <A|V> (A being the matrix and V the vector) and then find the range?

I hope you can help.

Thanks in advance.

 

[Dick]

You make sure (x,y,z,w)^T is in the span of the column vectors. If f is a definite linear transformation, it only has ONE range. You can't 'find a range' that would include an arbitrary vector unless you are allowed to change f. If you are allowed to change f, then as I said, change it in such a way that the vector is in the column space.

 

[Niles]

Thanks a lot to both of you.

 

[HallsofIvy]

I am not given a specific vector, but for this example we can equal it to (1,2,3,1) (just an example!). The text says "find the range of f(R^4) that has the above vector included".
The way you did it, you expressed the range in terms of that vector. I am quite sure that it is not what they are asking for. Are we supposed to write it in the form of <A|V> (A being the matrix and V the vector) and then find the range?

This, "find the range of f(R^4) that has the above vector included",still doesn't make sense to me. f has one range and either a given vector is in it or not. you can't "find" various ranges that have various vectors in it.

It still sounds to me that they are just telling you that you can use "x, y, z, and w" as the components of the vectors.