(3) (a) Show that every two-dimensional subspace of R3 is thekernel of some linear transformation T : R3 ? R. [Hint: there aremany possible ways to approach this problem. One is to use thefollowing fact, typically introduced in multivariable calculus: forevery plane P in R3, there are real numbers a, b, c, d such that apoint (x,y,z) belongs to P if and only if it satisfies the equationax+by+cz = d. You may use this fact without proof here, if youlike; note that it considers all planes, not just those through theorigin.] (b) Are there any other sets W such that W is the kernelof some linear transformation T : R3 ? R? (If not, explain why not;if so, explain why the set or sets you mention can be kernels, andwhy there are no others.) (c) What possibilities are there for theimage im(T) of a linear transformation T : R3 ? R? (d) Whatpossibilities are there for the kernel and image of a lineartransformation S : R ? R3?
