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exam1solutions - Math 217 Linear Algebra Fall 2002 Exam 1...

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Math 217, Linear Algebra, Fall 2002 Exam 1, October 4, 2002 Name: Solutions 1.(5pts) Let A be a 6 × 5 matrix. What must a and b be in order to define T : R a R b by T ( x ) = A x ? If we are trying to compute A x then x must be a length 5 vector. The result of A x is a length 6 vector. So a = 5 and b = 6. 2.(5pts) Give an example of a 2 × 2 matrix A which has the following three properties: 1) A 6 = 0 , 2) A 6 = I 2 , and 3) A T = A . Try 1 0 0 0 on for size. In fact, any matrix a b b c will work as long as if b = 0 then both a and d are not both 1 or both zero. 3.(10pts) Note that the matrix A = 4 5 6 7 8 7 6 5 6 7 8 9 is similar to the matrix B = 4 5 6 7 0 - 3 - 6 - 9 6 7 8 9 . Write down an elementary matrix E such that EA = B . What is E - 1 ? You can find an elementary matrix E such that EA = B by performing the same row operation on the identity as you did on A . That is, replace Row(2) of the identity with Row(2) - 2Row(1). Doing this yields the matrix E = 1 0 0 0 - 2 1 0 0 0 0 1 0 0 0 0 1 . The matrix E - 1 is such that E - 1 E = I 4 . So we find E - 1 by performing the same row operation on the identity as we would perform on E to obtain I 4 . That is, Row(2) Row(2)+2Row(1) transforms E to I 4 , thus Row(2) Row(2)+2Row(1) transforms I 4 to E - 1 = 1 0 0 0 2 1 0 0 0 0 1 0 0 0 0 1 . 4.(5pts) Consider the matrix A = 1 2 3 · · · 20 2 3 4 · · · 21 . . . . . . 19 20 21 · · · 39 . Are the columns of A linearly independent? (Explain).

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We have a theorem which states that a set of p vectors in R n for p > n is linearly dependent. Note that A has 20 columns and only 19 rows. That means that the 20 column vectors form a set of vectors R 19 , and we conclude that they must be linearly dependent.
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exam1solutions - Math 217 Linear Algebra Fall 2002 Exam 1...

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