Solve #15 Show every step and explain everything! POST PICTURES OF YOUR WORK and do not type it!
Solve #15 Show every step and explain everything! POST PICTURES OF YOUR WORK and do not type it!
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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Question
Solve #15 Show every step and explain everything! POST PICTURES OF YOUR WORK and do not type it!
Transcribed Image Text:vector v in v must be uniquely expressible as a linear
combination of the vectors in S.
(e) A set S of vectors in a vector space V spans V if and
only if the linear span of S is V.
(f) The linear span of two vectors in R³ must be a plane
through the origin.
(g) Every vector space V has a finite spanning set.
(h) If S is a spanning set for a vector space V, then any
proper subset S' of S (i.e., S' ‡ S) not a spanning set
for V.
(i) The vector space of 3 × 3 upper triangular matrices is
spanned by the matrices E¡¡ where 1 ≤ i ≤ j ≤ 3.
(j) A spanning set for the vector space P₂ (R) must contain
a polynomial of each degree 0,1, and 2.
(k) If m <n, then any spanning set for R" must contain
more vectors than any spanning set for Rm.
(1) The vector space P(R) of all polynomials with real
coefficients cannot be spanned by a finite set S.
Problems
For Problems 1-4, determine whether the given set of vectors
spans R2.
1. {(5, -1)}
2. {(1, 1), (2, 2), (2,3)}.
10. Show that the set of vectors
{(1, 2, 3), (3, 4, 5), (4, 5, 6)}
does not span R³, but that it does span the subspace
of R³ consisting of all vectors lying in the plane with
equation x - 2y + z = 0.
-
11. Show that v₁ = (2, −1), v₂ = (3, 2) span R² and ex-
press the vector v = (5, -7) as a linear combination
of V1, V₂.
=
12. Show that V₁ = (1,-5), V₂ (6, 3) span R², and
express the vector v = (x, y) as a linear combination
of V1, V2.
13. Show that V₁ = (1, −3, 2), v2 = (1, 0, −1), V3 =
(1, 2, –4) span R³, and express v = (9, 8, 7) as a
linear combination of V1, V2, V3.
=
14. Show that V₁
(-1, 3, 2), V₂ = (1, -2, 1), V3 =
(2, 1, 1) span R³, and express v = (x, y, z) as a linear
combination of V1, V2, V3.
15. Show that v₁ = (1, 1), V₂ = (-1,2), V3
=
R². Do V₁, V2 alone span R² also?
(1,4) span
16. Let S be the subspace of R³ consisting of all vectors
of the form v= (C₁, C2, C₂ - 2c₁). Determine a set of
vectors that spans S.
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