midterm 1 fall 2000

# Linear Algebra with Applications

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1 Mathematics 21b - Fall 2000 - First Exam This exam was 90 minutes long. Calculators were not allowed. 1) (12 points) True or False. (Circle one) You need not give your reasoning. a) The kernel of rref( A ) is the same as the kernel of A . a) TRUE FALSE b) The image of rref( A ) is the same as the image of A . b) TRUE FALSE c) Let A and B be n x n matrices, with AB = BA . Then A 3 B = BA 3 . c) TRUE FALSE d) There is a 4 x 4 matrix A such that image( A ) and kernel( A ) are the same subspace of R 4 . d) TRUE FALSE e) If A and B are n x n matrices such that the kernel of A is contained in the image of B , then the matrix AB cannot be invertible. e) TRUE FALSE f) If A is an invertible 2 x 2 matrix, then the rank of A is necessarily 1. f) TRUE FALSE 2) (18 points) Let A be the 4 x 5 matrix A = 1 2 1 5 0 1 2 1 1 0 3 6 1 7 1 1 2 0 3 0 - - - - - - - - . a) Represent the kernel of A as the span of a set of vectors. b) Represent the image of A as the span of a set of vectors. Eliminate any vectors which can be expressed as linear combinations of other vectors in your set. [This is called a basis for im(

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