Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.

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.

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
See similar textbooks
icon
Related questions
Question

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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS