MAS
Final-ex

A compute each of the following w 1 c w 1 b w 2 c w 2

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(a) Compute each of the following. [ w 1 ] C = [ w 1 ] B = [ w 2 ] C = [ w 2 ] B = [ w 3 ] C = [ w 3 ] B = (b) Obtain the matrix P. B C (c) What equation does P satisfy for all x in V ?? B C (d) Obtain a matrix Q which satisfies [ x ] C = Q[ x ] B for all x in the vector space V.

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Final Exam/MAS3105 Page 4 of 7 3. (20 pts.) Suppose matrices A and B are row equivalent, where 1 -3 4 -1 9 1 -3 0 5 -7 A = -2 6 -6 -1 -10 and B = 0 0 2 -3 8 -3 9 -6 -6 -3 0 0 0 0 5 3 -9 4 9 0 0 0 0 0 0 . Using this, solve the following problems. (a) (2 pts.) rank A = (b) (2 pts.) dim(Nul(A)) = (c) (3 pt.) Obtain a basis B 1 for Col(A). B 1 = (d) (3 pt.) Obtain a basis B 2 for Row(A). B 2 = (e) (3 pts.) Obtain a basis B 3 for Nul(A). (It might be wise to work on the back of page 3.) B 3 = (f) (4 pts.) Write each column of A which is not a member of B 1 as a linear combination of the elements of B 1 . (g) (3 pts.) Using your work from part (f) of this problem, write an arbitrary linear combination of the columns of A as a linear combination of the elements of B 1 .
Final Exam/MAS3105 Page 5 of 7 4. (10 pts.) (a) Starting with the basis B = {[1 1 0] T , [1 0 1] T , [0 1 1] T } for 3

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