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.

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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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solve #13, show all of your work and show step by step on PICTURES, DO NOT TYPE IT!

d L...
ofbede
ty
(45)
3. The
HOWY
HO
of
V
Differential Equations and Linear Algebra.pdf
Page 291 of 871
i
forms a subspace of V.
(b) If some vector v in a vector space V is a linear com-
bination of vectors in a set S, then S spans V.
(c) If S is a spanning set for a vector space V and W is a
subspace of V, then S is a spanning set for W.
(d) If S is a spanning set for a vector space V, then every
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)}.
M
8. {(1, 2, 3), (4, 5, 6), (7, 8, 9
V
Search
‚9)}.
9. Show that the set of vectors
{(−4, 1, 3), (5, 1, 6), (6, 0, 2)}
does not span R³, but that it does span the subspace
of R³ consisting of all vectors lying in the plane with
equation x + 13y - 3z = 0.
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.
(3, 2) span R² and ex-
11. Show that v₁ = (2, −1), V₂ =
press the vector v = (5, 7) as a linear combination
of V1, V₂2.
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 = (1, 4) span
R2. Do V₁, V2 alone span R² also?
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.
Transcribed Image Text:d L... ofbede ty (45) 3. The HOWY HO of V Differential Equations and Linear Algebra.pdf Page 291 of 871 i forms a subspace of V. (b) If some vector v in a vector space V is a linear com- bination of vectors in a set S, then S spans V. (c) If S is a spanning set for a vector space V and W is a subspace of V, then S is a spanning set for W. (d) If S is a spanning set for a vector space V, then every 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)}. M 8. {(1, 2, 3), (4, 5, 6), (7, 8, 9 V Search ‚9)}. 9. Show that the set of vectors {(−4, 1, 3), (5, 1, 6), (6, 0, 2)} does not span R³, but that it does span the subspace of R³ consisting of all vectors lying in the plane with equation x + 13y - 3z = 0. 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. (3, 2) span R² and ex- 11. Show that v₁ = (2, −1), V₂ = press the vector v = (5, 7) as a linear combination of V1, V₂2. 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 = (1, 4) span R2. Do V₁, V2 alone span R² also? 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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