Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - 1.3 Exercises - Page 34: 32

Answer

So the equation $x_1v_1+x_2v_2+x_3v_3=b$ has at least two solutions, not just one solution. (In fact, the equation has infinitely many solutions.)

Work Step by Step

See the parallelograms drawn on the figure from the text that accompanies this exercise. Here c1, c2, c3, and c4 are suitable scalars. The darker parallelogram shows that b is a linear combination of v1 and v2, that is $c_1v_1+c_2v_2+0\cdot v_3=b$ The larger parallelogram shows that b is a linear combination of v1 and v3, that is, $c_4v_1+0\cdot v_2+c_3v_3=b$ So the equation$ \ x_1v_1+x_2v_2+x_3v_3=b\ $ has at least two solutions, not just one solution. (In fact, the equation has infinitely many solutions.)
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