9. Suppose V1, V2, V3, V4 are linearly independent. Prove that the following vectors are also linearly independent - V1 V2, V2 - V3, V3 - V4, V4 10. Check if the following vectors are linearly independent or not: (a) (1, 1, 1), (1,2,3), (1, 5, 8) Є R³ (b) (1,2,0), (-1, 1, 2), (3, 0, −4) Є R³ (c) (−11,3,10), (1, −5, 26), (3, 4, −4), (1, 1, −1) Є R³ (d) sin(x), sin(2x) = C[-∞, ∞] (e) u = 2t23t+4, v = - (f) u = [1 1 [1] v = 2 2 2 " 4t2 [222] 333 3t+2 [333] w= ' 44 4 11. Determine if the vectors (1, 1, 1, 1), (1, 2, 3, 2), (2, 5, 6, 4), (2, 6, 8, 5) form a basis for R4. If not, find the dimension of the spanned subspace and basis for this subspace. (4.25)
9. Suppose V1, V2, V3, V4 are linearly independent. Prove that the following vectors are also linearly independent - V1 V2, V2 - V3, V3 - V4, V4 10. Check if the following vectors are linearly independent or not: (a) (1, 1, 1), (1,2,3), (1, 5, 8) Є R³ (b) (1,2,0), (-1, 1, 2), (3, 0, −4) Є R³ (c) (−11,3,10), (1, −5, 26), (3, 4, −4), (1, 1, −1) Є R³ (d) sin(x), sin(2x) = C[-∞, ∞] (e) u = 2t23t+4, v = - (f) u = [1 1 [1] v = 2 2 2 " 4t2 [222] 333 3t+2 [333] w= ' 44 4 11. Determine if the vectors (1, 1, 1, 1), (1, 2, 3, 2), (2, 5, 6, 4), (2, 6, 8, 5) form a basis for R4. If not, find the dimension of the spanned subspace and basis for this subspace. (4.25)
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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![9. Suppose V1, V2, V3, V4 are linearly independent. Prove that the following vectors are also linearly
independent
-
V1 V2, V2 - V3, V3 - V4, V4
10. Check if the following vectors are linearly independent or not:
(a) (1, 1, 1), (1,2,3), (1, 5, 8) Є R³
(b) (1,2,0), (-1, 1, 2), (3, 0, −4) Є R³
(c) (−11,3,10), (1, −5, 26), (3, 4, −4), (1, 1, −1) Є R³
(d) sin(x), sin(2x) = C[-∞, ∞]
(e)
u
=
2t23t+4, v =
-
(f) u =
[1 1 [1]
v =
2 2 2
"
4t2
[222]
333
3t+2
[333]
w=
'
44 4
11. Determine if the vectors (1, 1, 1, 1), (1, 2, 3, 2), (2, 5, 6, 4), (2, 6, 8, 5) form a basis for R4. If not, find
the dimension of the spanned subspace and basis for this subspace. (4.25)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F68065e9b-a71c-496a-bdfc-0d755e5e9bd8%2Fc250050f-3249-4e4f-aa4f-fe2b454896c9%2Fh0nw5dc_processed.png&w=3840&q=75)
Transcribed Image Text:9. Suppose V1, V2, V3, V4 are linearly independent. Prove that the following vectors are also linearly
independent
-
V1 V2, V2 - V3, V3 - V4, V4
10. Check if the following vectors are linearly independent or not:
(a) (1, 1, 1), (1,2,3), (1, 5, 8) Є R³
(b) (1,2,0), (-1, 1, 2), (3, 0, −4) Є R³
(c) (−11,3,10), (1, −5, 26), (3, 4, −4), (1, 1, −1) Є R³
(d) sin(x), sin(2x) = C[-∞, ∞]
(e)
u
=
2t23t+4, v =
-
(f) u =
[1 1 [1]
v =
2 2 2
"
4t2
[222]
333
3t+2
[333]
w=
'
44 4
11. Determine if the vectors (1, 1, 1, 1), (1, 2, 3, 2), (2, 5, 6, 4), (2, 6, 8, 5) form a basis for R4. If not, find
the dimension of the spanned subspace and basis for this subspace. (4.25)
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