Sample Quiz

April 5, 2006

Simplify the following expressions.

({{1, 7}, {3, -1}}) + ({{-1, 0}, {1, 0}}) =

({{2, 2, -7}, {3, 1, 4}}) + 3 ({{1, 0, 8}, {0, 1, 1}}) =

Det[({{1., 2.3}, {3., 7.}})] =

The matrix

M = ( {{1, 2, -1, 0, -3, 2, 21, 23}, {0, 0, 1, 2, 24, -1, 15, 17}, {1, 2, 2, 1, 24, 0, 28, 32}, {-1, -2, 0, 0, -3, 0, -4, -5}, {1, 2, 1, 0, 9, 0, 11, 13}} )

has reduced row echelon form

M∼ ({{1, 2, 0, 0, 3, 0, 4, 5}, {0, 0, 1, 0, 6, 0, 7, 8}, {0, 0, 0, 1, 9, 0, 10, 11}, {0, 0, 0, 0, 0, 1, 12, 13}, {0, 0, 0, 0, 0, 0, 0, 0}})

Give all solutions (in vector form) to the system of equations

x_1 + 2 x_2 - x_3 - 3 x_5 + 2 x_6 + 21 x_7 = 23

x_3 + 2 x_4 + 24 x_5 - x_6 + 15 x_7 = 17

x_1 + 2 x_2 + 2 x_3 + x_4 + 24 x_5 + 28 x_7 = 32

-x_1 - 2 x_2 - 3 x_5 - 4 x_7 = -5

x_1 + 2 x_2 + x_3 + 9 x_5 + 11 x_7 = 13

({{x_1}, {x_2}, {x_3}, {x_4}, {x_5}, {x_6}, {x_7}}) =

Are the rows of M linearly independent?

What is the dimension of the span of the rows of M?

Give a basis for the span of the rows of M.

What is the reduced row echelon form of the matrix

R = ( {{1, 2, 0, -3, 2, 23}, {0, 0, 2, 24, -1, 17}, {1, 2, 1, 24, 0, 32}, {-1, -2, 0, -3, 0, -5}, {1, 2, 0, 9, 0, 13}} ) ?

Hint: you don't need to do much computation to answer this!

Are the vectors (1,2), (3,-3) linearly independent?

Are the vectors (1,2), (1,3), (1,4) linearly independent?

Solutions

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