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67. Consider linearly independent vectors v1, v2, ..., vp a subspace V of R" and vectors w1, w2, ..., ūwg that span V. Show that there is a basis of V that consists of all the v; and some of the w j. Hint: Find a basis of the image of the matrix in | A = | 01 W1 p .. .. | | 68. Use Exercise 67 to construct a basis of R* that consists of the vectors 1. [1' 4 3 4 8. and some of the vectors ē1, ė2, é3, and ē4 in R“.

67. Consider linearly independent vectors v1, v2, ..., vp a subspace V of R" and vectors w1, w2, ..., ūwg that span V. Show that there is a basis of V that consists of all the v; and some of the w j. Hint: Find a basis of the image of the matrix in | A = | 01 W1 p .. .. | | 68. Use Exercise 67 to construct a basis of R* that consists of the vectors 1. [1' 4 3 4 8. and some of the vectors ē1, ė2, é3, and ē4 in R“.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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67. Consider linearly independent vectors v1, v2, ..., ip in a subspace V of R' and vectors w1, span V. Show that there is a basis of V that consists of all the v; and some of the wj. Hint: Find a basis of the image of the matrix w2, ..., wg that | | A = Wi .. .. | | 68. Use Exercise 67 to construct a basis of R4 that consists of the vectors 1 2 4 3 6. 4 8 and some of the vectors e1, é2, é3, and é4 in R*.
Transcribed Image Text:67. Consider linearly independent vectors v1, v2, ..., ip in a subspace V of R' and vectors w1, span V. Show that there is a basis of V that consists of all the v; and some of the wj. Hint: Find a basis of the image of the matrix w2, ..., wg that | | A = Wi .. .. | | 68. Use Exercise 67 to construct a basis of R4 that consists of the vectors 1 2 4 3 6. 4 8 and some of the vectors e1, é2, é3, and é4 in R*.
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