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midterm_07

# midterm_07 - C = £ 4 5 6 1 2 3 = D = 1 2 3 £ 4 5 6 =(b...

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14 February 2007 MATH 133 Mid-term Time: 50 minutes Instructions: This question sheet cannot be taken from the exam room. Answer all questions in the exam booklets. No calculators, cell-phones, or other electronic devices allowed. No books or notes allowed. 1. (5 points) Let P = (1 , 2 , 3) and Q = (7 , 14 , 21). Let M be the midpoint of the line segment PQ and R and S be the points that divide PM ( not PQ ) into three equal parts. Sketch the situation and find M , R and S . 2. (5 points) (a) Define briefly the three types of elementary row operations on a matrix. (b) Using elementary row operations, find the reduced echelon form of the following matrix. Show each step - write a new matrix showing the result of each of your elementary row operations. 0 1 1 4 1 1 2 - 1 3 4 7 1 3. (5 points) (a) Evaluate each of the following matrix products: C = £ 4
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Unformatted text preview: C = £ 4 5 6 / 1 2 3 = ? , D = 1 2 3 £ 4 5 6 / = ? (b) Find rank( C ) and rank( D ), justifying your answers. 4. (10 points) Let A = 1-2 2 2-3 6-6-6 0-1 2 2 6 4 1-2 3 4 1 , B = 1-2 0-2-2 1 2 1 . Given that B is the reduced echelon form of A , answer the following: (a) Find the dimension of the row space of A and a basis of it. (b) Find the dimension of the column space of A and a basis of it. (c) Find the dimension of the null space of A and a basis of it. (d) Is it true or false that col( A ) = Span ( -2 2 , -2 1 ) ? Why do you say so ? 1...
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