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2. (a) Explain what is mean by a linearly independent set of vectors. For which values of k is the following set of vectors linearly independent? {(1, 3, 3, 1), (2, 0, 1, 9), (0, 0, 4, 3), (k, k, −4,4)} (b) What is meant by a basis for a subspace of R"? Find bases for each of the following subspaces of R³ (there is no need to show that they are subspaces): (i) {(a 2b, a + 3b, 4b): a, b ≤ R}; (ii) {(x, y, z) = R³ : x - 2y - 6z = 0}.

2. (a) Explain what is mean by a linearly independent set of vectors. For which values of k is the following set of vectors linearly independent? {(1, 3, 3, 1), (2, 0, 1, 9), (0, 0, 4, 3), (k, k, −4,4)} (b) What is meant by a basis for a subspace of R"? Find bases for each of the following subspaces of R³ (there is no need to show that they are subspaces): (i) {(a 2b, a + 3b, 4b): a, b ≤ R}; (ii) {(x, y, z) = R³ : x - 2y - 6z = 0}.

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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2. (a) Explain what is mean by a linearly independent set of vectors. For which values of k is the following set of vectors linearly independent? {(1, 3, 3, 1), (2, 0, 1, 9), (0, 0, 4, 3), (k, k, -4,4)} (b) What is meant by a basis for a subspace of R"? Find bases for each of the following subspaces of R³ (there is no need to show that they are subspaces): (i) {(a − 2b, a + 3b, 4b) : a, b ≤ R}; (ii) {(x, y, z) = R³ : x - 2y - 6z = 0}.
Transcribed Image Text:2. (a) Explain what is mean by a linearly independent set of vectors. For which values of k is the following set of vectors linearly independent? {(1, 3, 3, 1), (2, 0, 1, 9), (0, 0, 4, 3), (k, k, -4,4)} (b) What is meant by a basis for a subspace of R"? Find bases for each of the following subspaces of R³ (there is no need to show that they are subspaces): (i) {(a − 2b, a + 3b, 4b) : a, b ≤ R}; (ii) {(x, y, z) = R³ : x - 2y - 6z = 0}.
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