Matrix Representation for Linear Transformation

Suppose: T: R^{3} ----> R^{3} is a linear transformation whose kernel has dimension 2. Describe the form of a matrix representation for T.

We did numerous examples of linear transformations from R^{3} ----> R^{3} Doesn't this mean that the matrix has to be 3x3 since they all were in my class? If the kernel has dimension 2, the image has dimension 1. So isn't the only possibility of a matrix representation for T a 3x3?

Any help would be great.

Re: Matrix Representation for Linear Transformation

Hey TimsBobby2.

For this to be 2-dimensional it means that you have one vector being a linear combination of the other two vectors.

So if you fix two vectors in the matrix, then make the other one a linear combination of the others.

Re: Matrix Representation for Linear Transformation

Quote:

Originally Posted by

**TimsBobby2** Suppose: T: R^{3} ----> R^{3} is a linear transformation whose kernel has dimension 2. Describe the form of a matrix representation for T.

We did numerous examples of linear transformations from R^{3} ----> R^{3} Doesn't this mean that the matrix has to be 3x3 since they all were in my class? If the kernel has dimension 2, the image has dimension 1. So isn't the only possibility of a matrix representation for T a 3x3?

Any help would be great.

Yes, the matrix representation of a linear transformation from $\displaystyle R^3$ to $\displaystyle R^3$ must be a 3 by 3 matrix. But that isn't the question! The matrix representation also depends upon what basis vectors you use. If the kernel is 2 dimensional, then you can choose two independent vectors from the basis and a third vector, independent of the first two, (in fact, perpendicular to the first two) as basis vectors. What would the matrix look like in that case?