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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
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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Solve #15  Show every step and explain everything! POST PICTURES OF YOUR WORK and do not type it!

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.
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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