math 235 hw1 - Homework 1 Due February 2 Adam Gamzon...

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Unformatted text preview: Homework 1, Due February 2 Adam Gamzon Related reading: sections 1.1 - 1.3 from the text, handout titled ”Matrix Formalism” 1. Consider the system x 1- 7 x 2 + x 5 = 3 , x 3- 2 x 5 = 2 , x 4 + x 5 = 1 . (a) Find the augmented matrix for this system. Solution: The augmented matrix is 1- 7 0 0 1 | 3 1 0- 2 | 2 0 1 1 | 1 . (b) Use Gauss-Jordan elimination to find all solutions to this equation. Write the solutions in vector form. Solution: The matrix from part (a) is already in rref. From this, we see that x 2 and x 5 are free variables. Set x 2 = t and x 5 = s where s and t are arbitrary real numbers. Then we can write every solution in the form x 1 x 2 x 3 x 4 x 5 = 7 t- s + 3 t 2 s + 2- s + 1 s = t 7 1 + s - 1 2- 1 1 + 3 2 1 . 2. Find all solutions of the equation - 8- 1 2 15 = x 1 1 4 7 5 + x 2 2 5 8 3 + x 3 4 6 9 1 . (Hint: rewrite this as a matrix equation.) Solution: Finding all solutions to this equation is the same as finding all solutions to the equation 1 2 4 4 5 6 7 8 9 5 3 1 x 1 x 2 x 3 = - 8- 1 2 15 ....
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This note was uploaded on 11/22/2011 for the course MATH 255 taught by Professor Staff during the Spring '10 term at UMass (Amherst).

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math 235 hw1 - Homework 1 Due February 2 Adam Gamzon...

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