sol2 - . 2) (5 points) Let B = 1 2 2-5 1 2 3-7-2-4-6 14 ....

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Mathematics 20F Fall 2008 Name: Quiz 2 No aids are allowed. Be sure to write down all row operations used. There are two questions on this quiz, one on this side, the other on the fip side. 1) (5 points) Let A be the matrix given below whose reduced row echelon form is R , also given below. Compute bases for Col( A ) and Nul( A ). A = 2 4 8 2 4 1 4 8 4 5 - 1 - 4 - 8 - 5 - 6 R = 1 0 0 0 1 0 1 2 0 0 0 0 0 1 1
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For a basis for Col( A ), we will choose the pivot columns of A . To do so, we identify that columns 1, 2, and 4 are the pivot columns using the reduced row echelon form. Thus, we have 2 1 - 1 , 4 4 - 4 , 2 4 - 5 . as a basis for Col( A ). To ±nd a basis for Nul( A ), we use the reduced row echelon form and solve R x = 0 . Here, we have x = x 3 0 - 2 1 0 0 + x 5 - 1 0 0 - 1 1 . Thus, a basis for Nul( A ) is 0 - 2 1 0 0 , - 1 0 0 - 1 1
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Unformatted text preview: . 2) (5 points) Let B = 1 2 2-5 1 2 3-7-2-4-6 14 . Compute rank( B ) and dim Nul( B ). In order to compute rank( B ), we must identify the number of pivot columns of B . Thus, we begin by row reducing: 1 2 2-5 1 2 3-7-2-4-6 14 R 2 R 2-R 1------- 1 2 2-5 1-2-2-4-6 14 R 3 R 3 +2 R 1------- 1 2 2-5 0 0 1-2 0 0-2 4 R 3 R 3 +2 R 2------- 1 2 2-5 0 0 1-2 0 0 0 . From this row echelon form, we can identify that the matrix has 2 pivot columns. Thus, rank( B ) = 2. Now, using the identity that for an m n matrix, n = rank( B ) + dim Nul( B ), in this case, we have dim Nul( B ) = 4-2 = 2....
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sol2 - . 2) (5 points) Let B = 1 2 2-5 1 2 3-7-2-4-6 14 ....

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