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301_F10_hw3

301_F10_hw3 - (e What is the dimension of the column space...

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AME 301 Homework #3 | Due Mon Sep 20, 2010 1. Consider the following system of equations: 2 4 1 2 3 4 3 6 10 14 ± 2 ± 4 ± 9 ± 14 3 5 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = 2 4 b 1 1 1 3 5 (a) Simplify the system by applying row reduction to the augmented matrix. (b) Does the solution exist for all values of b 1 ? If not, determine the value of b 1 which allows a solution. (c) From the row-reduced form, identify the basic variables and free variables. Using the value of b 1 found in part (b), obtain the solution by setting the free variables to arbitrary constants and solving for the basic variables. (d) From the solution obtained in part (c), determine the dimension of the null space of A , and a basis for the null space.
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Unformatted text preview: (e) What is the dimension of the column space of A ? By noting the basic variables, determine a basis for the column space of A without row-reducing A T . (f) If b lies in the column space of A , it should be expressible as a linear combination of the basis vectors found in part(e), i.e. b = c 1 v 1 + ² ² ² . By inspection of the solution found in part (c), determine the coe±cients that appear in this linear combination, and verify that the linear combination of basis vectors indeed sums to the vector b used in part (c). 2. (Additional problems will be sent later) 1...
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