MATH 33AH Study Guide - Final Guide: Invertible Matrix, Linear Map, Solution Set

Let A = , b = Find all vectors x R3 such that Ax = 0 Recall that 0 is the zero vector whose entries are all zeroes. Find all vectors x R3 such that Ax = b Denote by ker(A) = {x R3|Ax = 0}. Show that if x, y ker(A), and alpha R, then x + y ker(A), and alpha x ker(A). The set ker(A) is called the kernel of the matrix A. The set of vectors b such that Ax = b for some x is called the image of A.

4. Define T: R R' by Tx) - AX where A--1 -2 0 Find ker(T). Verify the Rank/Nullity Theorem (Theorem 3). Is T one-to-one 2 4 1 [3 marks] hat valuefs) of k if any, is the transformation T(x, y) 3y+ k linear? 13 marks]

Please explain the answer provided below in such a way as to be understood by someone who is really struggling in linear algebra. THANKS!!! Let N be a 2x2 complex matrix such that N2 = 0. Prove that either N = 0 or N is similar over C to Letf: C2 rightarrow C2 be the linear mapping defined by f(x) = Nx. Then f(f(x)) = N(Nx) = 0 for allx. Hence the image off is a subspace of the kernel off. If dim im(f) = 2, then f is bijective and fLambda2 cannot be the zero mapping, contradiction. Hence dim im(f) is 0 or 1. If dim im(f) = 0, then f is the zero mapping and N is the zero matrix. If dim im(f) = 1, then dim ker(f) = 1, and hence im(f)=ker(f). Denote this subspace by V. Then there is a vector y in C2 which is not in V. Then f(y) is in V and is not zero, hence {y, f(y)} is a basis. With respect to this basis, the matrix representation off is [00; 10]. Hence N is similar to the matrix [0 0; 1 0].