8. Decide which of the following sets are linearly independent in Rn. Justify your answer in each case. (a) X₁ {(1,0), (0, 1)} ℃ R² (b) X₂= {(1,0), (2,0)} CR² (c) X3 = {(-1,0), (0, 0)} CR² (d) X4 = {(1,0), (0, 1), (1, 1)} CR² (e) X5 = {(1,0,0), (0, 1, 0), (1, 1, 1)} C R³ (f) X6 = {(1,0,0), (0, 1,0), (1, 1, 1), (-1,0,1)} CR³ (g) X7 = {(0,0,0), (0, 1,0), (1, 1, 1), (-1,0,1)} CR³ (h) Xs = {(0,1,0), (1, 1, 1)} CR³ (i) X9 = {(4,3,0,0), (0, 0, 1, 1), (0, 0, 0, 1), (1, 0, 0, 1), (0, 1, 0, 1)} ℃ R4 (j) X10 = {(1,2,0, 0), (0, 2, 3,0), (0, 0,-1,1)} CR¹ 9. Decide, for the sets X¿ above, for which i, j = {1, 2, .. 2 10}, span(X₂) = span(X₂).
8. Decide which of the following sets are linearly independent in Rn. Justify your answer in each case. (a) X₁ {(1,0), (0, 1)} ℃ R² (b) X₂= {(1,0), (2,0)} CR² (c) X3 = {(-1,0), (0, 0)} CR² (d) X4 = {(1,0), (0, 1), (1, 1)} CR² (e) X5 = {(1,0,0), (0, 1, 0), (1, 1, 1)} C R³ (f) X6 = {(1,0,0), (0, 1,0), (1, 1, 1), (-1,0,1)} CR³ (g) X7 = {(0,0,0), (0, 1,0), (1, 1, 1), (-1,0,1)} CR³ (h) Xs = {(0,1,0), (1, 1, 1)} CR³ (i) X9 = {(4,3,0,0), (0, 0, 1, 1), (0, 0, 0, 1), (1, 0, 0, 1), (0, 1, 0, 1)} ℃ R4 (j) X10 = {(1,2,0, 0), (0, 2, 3,0), (0, 0,-1,1)} CR¹ 9. Decide, for the sets X¿ above, for which i, j = {1, 2, .. 2 10}, span(X₂) = span(X₂).
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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Can you answer Question 9 Decide, for the sets Xi above, for which i, j ∈ {1, 2, · · · , 10}, span(Xi) = span(Xj ).
Transcribed Image Text:8. Decide which of the following sets are linearly independent in R". Justify your
answer in each case.
(a) X₁ = = {(1,0), (0, 1)} ℃R²
(b) X₂ = {(1,0), (2,0)} CR²
(c) X3 =
{(-1,0), (0,0)} ℃ R²
(d) X4 =
{(1,0), (0, 1), (1, 1)} C R²
(e) X5 = {(1,0,0), (0, 1, 0), (1, 1, 1)} CR³
(f) X6 = {(1,0,0), (0, 1, 0), (1, 1, 1), (−1, 0, 1)} ℃ R³
(g) X7 = {(0,0,0), (0, 1, 0), (1, 1, 1), (–1, 0, 1)} ℃ R³
(h) X8 = {(0, 1,0), (1, 1, 1)} C R³
(i) X9 = {(4, 3, 0, 0), (0, 0, 1, 1), (0, 0, 0, 1), (1, 0, 0, 1), (0, 1, 0, 1)} ℃ R4
(j) X10 = {(1, 2, 0, 0), (0, 2, 3, 0), (0, 0, −1, 1)} ℃ Rª
9. Decide, for the sets X; above, for which i, j = {1, 2, ..., 10}, span(X₂) = span(X;).
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