16.Find a basis and dimension of the subspace W of Rª spanned by(i) (1,4, -1, 3), (2, 1, –3, –1) and (0, 2, 1, -5)(ii) (1, –4, –2, 1); (1, –3, –1, 2) and (3, –8, –2, 7).

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Answered: 16. Find a basis and dimension of the…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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a) Verify the given sequence B is a basis ; and b) compute the images of the standard basis vectors with respect to the given linear transformation. 15. fi = 1+2x – 2x2, f2 = 2 + 3x – 4x2, f3 = 1 + 4x – x², B = (f1, f2, f3). - 1 T(fi) = (; 10)- 4 T(fi) = (; ).T ,T(f2) : 3 8
Consider the subspace S of the Euclidean inner product space R+ spanned by the vectors v₁ = (1,1,1,1), v₂=(1,1,2,4), v₂=(1,2,-4,-3). Find an orthogonal basis of S. O*((-.-3.1).(1.2.-4,-3).(4,2,1,1)) OB. © ³ {(1,1,2,4), ( − 1, − 1,0,2 A. -1,0,2), (12. 2.-3,1)} O C. None in the given list. OD. {(1,1,1,1),(1,1,2,4), (1,2,-4,-3) } OE {(1,1,1,1),(-1,-1,0,2), (1,3, -6,2) }
2. Let x = 4 (19) 10 and H be the subspace of R³ with basis B Find [x]B. -{Q).()} 5 5 =
(b) If By = {v1, V2, . .. , V,} is a basis for V, show that there exist n +1 nested subspaces of V.
12. Find the dimension of the subspace of R spanned by the following set of vectors. {(1, 5, 6), (2, 6, 8), (3, 7, – 1), (4, 8, 12)] (a) 1 (b) 2 (c) 3 (d) 4 (e) 0
4) Find a basis for the subspace of R* spanned by S. S = {(2,5,-3, -2), (-2, -3, 2, -5)(1,3,-2, 2)(-1,-5,3,5)}.
Let L: R3 R2 be defined by the following. X1 + X3 ***] 8x2 Suppose X1 L X₂ P = = 1 -(HH) 4 3 3 1 2 B₁ = 4 ={[8][;}} is an ordered basis for the domain and B₂ = is an ordered basis for the range. Find the matrix representation P for L relative to B₁ and B₂ such that [L(u)]b₂ = P[u]ß₁'
18. Find a basis and dimension of the orthogonal complement to the span ((1, 2, 0, -1), (0, 1, 1, 0), (1, 1, −1, −1)).
3. Given A= [1 0 (a) Give the vector c so that Ax = b has a solution if and only if b - c = 0. (b) Give a basis of the subspace of all vectors b such that Ax = b has a solution. %3D
Suppose V is a subspace of R" and suppose {v1, v2, v3} is a basis of V. Decide if the following sets of vectors are a basis for V: (i) {v2, v1 – 503, 2v3} (ii) {v2, v1 – 503, 203, 302 + 703 – v1} (iii) {202 – v3, v1}
10. (Strang 4.1.28) Explain why each of these statements is false: (a) ā = [1, 1, 1]" is perpendicular to b = [1, 1, –2]", so the planes a7 = x + y + z = 0 and b"r = x + y – 2z = 0 are orthogonal subspaces of R³. (b) The subspace of R spanned by (1, 1, 0, 0, 0) and (0,0,0, 1, 1) is the orthogonal com- plement of the subspace spanned by (1, – 1,0, 0,0) and (2, –2, 3, 4, –4). (c) Two subspaces that meet only at the zero vector must be orthogonal.

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16. Find a basis and dimension of the subspace W of Rª spanned by (i) (1, 4. –1, 3)» (2, 1, –3, –1) and (0. 2, 1, –5) (ii) (1, –4, –2, 1); (1, –3, –1, 2) and (3, –8, –2, 7).