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# 2 - Assignment for Applied Linear Algebra Math 270 The four...

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Assignment for Applied Linear Algebra - Math 270 January 28, 2011 The four exercises amount to 15 points. Obtaining 13 points out of 15 guarantees full marks. The deadline is set to February 14. (Please see details in the Outline of the course, on WebCT.) Marks. Just sum the points, no % or letters. Problem 1 (2 pts.) Show that u 1 = (1 , 1 , 1) T , u 2 = (1 , - 1 , 0) T , u 3 = (1 , 1 , - 2) T , form a basis of R 3 . Write v = (1 , - 2 , 5) T in the coordinates of the basis { u 1 , u 2 , u 3 } . SOLUTION: Three vectors in R 3 are a basis if and only if they are linearly independent. You can check this for example by computing det 1 1 1 1 - 1 1 1 0 - 2 = 6 6 = 0 , or showing that the system 1 1 1 1 - 1 1 1 0 - 2 x 1 x 2 x 3 = 0 0 0 has a unique solution, which means that the unique linear combination of u 1 , u 2 , u 3 is the one where all coefficients are zero. (1 pt.) Solving 1 1 1 1 - 1 1 1 0 - 2 x 1 x 2 x 3 = 1 - 2 5 gives the coordinates of v w.r.t. the basis { u i } . The coordinates are [4 / 3 , 3 / 2 , - 11 / 6] . (1 pt.) Problem 2 (3 pts.) Recall the definition of linear subspace of a vector space.

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2 - Assignment for Applied Linear Algebra Math 270 The four...

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