(1) Let T: Rn--->Rm be linear tranformations.

a. If T maps Rnonto Rm, give a relationship between m and n

b. If T is one-to-one, give a relationship between m and n

c. If T maps Rn onto Rm and is one-to-one, give a relationship between m and n

(Hint: Think about the size of the standard matrix representation of T and the placement of the pivots in each case)

(2) Let T: R3 ---> R4 be a linear transformation such that the only solution to T(x) = 0 is trivial solution.

a. If T is one-to-one

b. Does T map R3onto R4?

Justify your answers in each case.

(Hint:one way to approach this is to look at what the martix representation of T might look like and where it does or does not have pivots.)

(3) Suppose a linear transformation T: R2> R2 is formed by taking a rotation counterclockwise of 90 degrees, follwed by a reflection through the X2-axis. Describe the points that will be moved back to their original position by this transformation?

(Hint: Think about what T will do to the unit box and the vectors e1 and e2)