Thread: Linear transformations problem (lin. algebra)

The following problem is confusing me so any help would be greatly appreciated!

Find the set of all matrices with respect to the standard basis for R^2 for all linear operators that take all vectors of the form [0, y] to vectors of the form [0, y'].

Originally Posted by buckaroobill

The following problem is confusing me so any help would be greatly appreciated!

Find the set of all matrices with respect to the standard basis for R^2 for all linear operators that take all vectors of the form [0, y] to vectors of the form [0, y'].

Let such a matric be A then:

A (0,y)' = (A_{1,2}y, A_{2,2}y)'

so if A (0, y) = (0, y') for some y', then A_{1,2}=0. Also if A_{1,2}=0 then
A (0, y) = (0, y') for some y'.

So the set of 2x2 matrices you seek is the set with the right hand top
corner element equal to zero.

(This is when we consider the matrix acting on a column vector, if we have
row vectors and the action is in the order yA, then the A will be the transpose
of that in the other representation of the linear operator and the set will be
the set with the left hand bottom corner element equal to zero.)

RonL