(3) (a) Show that every two-dimensional subspace of R3 is the kernel of some linear transformation...

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

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Colby Messinger answered 1 year ago

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