Question

determine whether multiplication by A is one to one linear transformation
a)A=[ 1 -1
2 0
3 -4]
b) A=[ 1 2 3
-1 0 4]
is there another way besides kernell one?

Answer 1

You're probably on the right track. The kernel of the transformation corresponds to the null space of the matrix. What do you know about the null spaces for each of your matrices in A and B above? The general solution is x_gen = x_particular + x_nullspace. If there are free variables we have infinite solutions from the null space (i.e. solutions of Ax=0). In this case is a one to one mapping possible? i.e. can you get back to the 'x' you started with after having multiplied it by Ax? In order for there to be a solution at all, you must be able to solve for the particular solution (Ax=b). If there is a particular solution and only the zero vector in the null space, then there's a unique solution. What does that tell you about the number of free variables of A and its column rank?

Answer 2

hump day woot woot.