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7. Determine whether each of the following sets of vectors forms a basis for R³. If a set does not form a basis, add and/or remove vectors to the set so that the result is a basis for R³. For each set, add and remove the fewest number of vectors possible. (a) {(0, log(2), 0), (π, 0, 0), (0, 0, 1)} (b) {(1, 0, 0), (0, 2, 3)} (c) {(3, 4, 1), (2, 1, 1), (-4, -2,-2)} (d) {(1, 1, 1), (0, -2,-2), (0, 0, 8), (1, 2, 3)}

7. Determine whether each of the following sets of vectors forms a basis for R³. If a set does not form a basis, add and/or remove vectors to the set so that the result is a basis for R³. For each set, add and remove the fewest number of vectors possible. (a) {(0, log(2), 0), (π, 0, 0), (0, 0, 1)} (b) {(1, 0, 0), (0, 2, 3)} (c) {(3, 4, 1), (2, 1, 1), (-4, -2,-2)} (d) {(1, 1, 1), (0, -2,-2), (0, 0, 8), (1, 2, 3)}

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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7. Determine whether each of the following sets of vectors forms a basis for R³. If a set does not form a basis, add and/or remove vectors to the set so that the result is a basis for R³. For each set, add and remove the fewest number of vectors possible. (a) {(0, log(2), 0), (π, 0, 0), (0, 0, 1)} (b) {(1,0,0), (0, 2, 3)} (c) {(3, 4, 1), (2, 1, 1), (—4, —2, −2)} (d) {(1, 1, 1), (0, -2,-2), (0, 0, 8), (1,2,3)}
Transcribed Image Text:7. Determine whether each of the following sets of vectors forms a basis for R³. If a set does not form a basis, add and/or remove vectors to the set so that the result is a basis for R³. For each set, add and remove the fewest number of vectors possible. (a) {(0, log(2), 0), (π, 0, 0), (0, 0, 1)} (b) {(1,0,0), (0, 2, 3)} (c) {(3, 4, 1), (2, 1, 1), (—4, —2, −2)} (d) {(1, 1, 1), (0, -2,-2), (0, 0, 8), (1,2,3)}
Expert Solution
Step 1

The given vectors are,

a) 0, log2, 0, π, 0, 0, 0, 0, 1

b) 1, 0, 0, 0, 2, 3

c) 3, 4, 1, 2, 1, 1, -4, -2, -2

d) 1, 1, 1, 0, -2, -2, 0, 0, 8, 1, 2, 3.

We have to find these vectors to form a basis of ℝ3. If not, add or remove some vectors to make a basis.

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