samplemidtermsolutions

samplemidtermsolutions - Sample Midterm 1, Math 33A, Fall...

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Unformatted text preview: Sample Midterm 1, Math 33A, Fall 2007 October 23, 2007 1 1. Indicate whether each statement is true or false. (You need not show your work.) (2 pts) There exists a 3 by 3 matrix A such that imageA = ker A . T F (2 pts) If A and B are n by m matrices, and the vector ~v is in imageA and imageB, then it must be in the image of the matrix ( A + B ) as well. T F (2 pts) If A is the 2 by 2 matrix which represents counterclockwise rotation by an angle 2 , and B is the 2 by 2 matrix which represents reflection across the x 1-axis, then AB = BA . T F (2 pts) If A is a n by m matrix and the columns of A are linearly independent, then imageA must be all of R n . T F (2 pts) If the matrix A is invertible, then A 2 must be invertible as well. T F Solution. F, F, F, F, T. If A is a 3 by 3 matrix, then by the rank-nullity theorem, dim ker A +dim imageA = 3. If ker A = imageA, they would have the same dimension, which is impossible. If A = I n and B =- I n , then both A and B have image equal to all of R n . Since A + B = 0, image(A + B) = { ~ } ....
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samplemidtermsolutions - Sample Midterm 1, Math 33A, Fall...

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