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midterm_2_sample

# midterm_2_sample - b Find a basis for ker A What is the...

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MA 35100 GOINS SAMPLE MIDTERM EXAM #2 This exam is to be done in 50 minutes in one continuous sitting. Collaboration is not allowed, but you may use your own personal notes; the lecture notes posted on the course web site; homework solutions posted on the course web site; and Bretscher’s Linear Algebra with Applications . Write your answers on a separate sheet of paper. Provide all details of your work : either give precise references for or proofs of any statements you claim. Problem 1. Consider the matrix A = 1 1 1 1 2 5 1 3 9 . a. Compute its reduced row-echelon form. Show all of your work.
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Unformatted text preview: b. Find a basis for ker( A ). What is the nullity of A ? c. Find a basis for im( A ). What is the rank of A ? Problem 2. Let A and B be n × n matrices. a. Show that ker( A ) ⊆ ker( B A ). b. How does the rank of A compare to the rank of B A ? Explain using the Rank-Nullity Theorem. Problem 3. Let B = ( ~v 1 ,~v 2 ) be the basis for R 2 in terms of the vectors ~v 1 = ± 1 ² and ~v 2 = ±-1 ² . For a scalar k , consider the linear transformation T : R 2 → R 2 deﬁned by T ( ~x ) = ± 1 k 0 1 ² ~x. Compute the B-matrix of T . 1...
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