span 0. Show that R-. 1 11. Show that R3 span 12. Show that R3 span In Exercises 13-16, describe the span of the given vectors (a) geometrically and (b) algebraically. 2 13. 3 14. 3 15. 2 2 16. 2 17. The general equation of the plane that contains the points (1, 0, 3), (-1, 1, -3), and the origin is of the form ax by +cz = 0. Solve for a, b, and c. 2 18. Prove that u, v, and w are all in span(u, v, w). 19. Prove that u, v, and w are all in span(u, u+ v, u + v+w) 20. (a) Prove that if u,.. , are vectors in R", S = u, u2,.. ., u}, and T = {u1, . . , U uk+1. . un n, then span(S) C span(T). [Hint: Rephrase this question in terms of linear combinations.] (b) Deduce that if R" = span(S), then R" = span ( T) 30 also. 31. 21. (a) Suppose that vector w is a linear combination of vectors u, . . , u and that each u, is a linear combination of vectors v, . . , V Prove that w is a linear combination of v,.. . , Vm and therefore span(u,,..., uk) span(vi,..,v,,) In E give

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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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span
0. Show that R-.
1
11. Show that R3
span
12. Show that R3
span
In Exercises 13-16, describe the span of the given vectors
(a) geometrically and (b) algebraically.
2
13.
3
14.
3
15. 2
2
16.
2
17. The general equation of the plane that contains the
points (1, 0, 3), (-1, 1, -3), and the origin is of the
form ax by +cz = 0. Solve for a, b, and c.
2
18. Prove that u, v, and w are all in span(u, v, w).
19. Prove that u, v, and w are all in span(u, u+ v, u +
v+w)
20. (a) Prove that if u,..
, are vectors in R", S =
u, u2,.. ., u}, and T = {u1, . . , U uk+1. .
un n, then span(S) C span(T). [Hint: Rephrase this
question in terms of linear combinations.]
(b) Deduce that if R" = span(S), then R" = span ( T)
30
also.
31.
21. (a) Suppose that vector w is a linear combination
of vectors u, . . , u and that each u, is a linear
combination of vectors v, . . , V Prove that w is
a linear combination of v,.. . , Vm and therefore
span(u,,..., uk) span(vi,..,v,,)
In E
give
Transcribed Image Text:span 0. Show that R-. 1 11. Show that R3 span 12. Show that R3 span In Exercises 13-16, describe the span of the given vectors (a) geometrically and (b) algebraically. 2 13. 3 14. 3 15. 2 2 16. 2 17. The general equation of the plane that contains the points (1, 0, 3), (-1, 1, -3), and the origin is of the form ax by +cz = 0. Solve for a, b, and c. 2 18. Prove that u, v, and w are all in span(u, v, w). 19. Prove that u, v, and w are all in span(u, u+ v, u + v+w) 20. (a) Prove that if u,.. , are vectors in R", S = u, u2,.. ., u}, and T = {u1, . . , U uk+1. . un n, then span(S) C span(T). [Hint: Rephrase this question in terms of linear combinations.] (b) Deduce that if R" = span(S), then R" = span ( T) 30 also. 31. 21. (a) Suppose that vector w is a linear combination of vectors u, . . , u and that each u, is a linear combination of vectors v, . . , V Prove that w is a linear combination of v,.. . , Vm and therefore span(u,,..., uk) span(vi,..,v,,) In E give
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