1. Let S = {(1, 0, -1, -1),(1, -1, 1, 2), (5, 2, -9, -11)} CR¹.a) Show that S' is linearly dependent over R.b) Determine a basis of Span (S) and dim (Span (S))R4c) Find a basis of Rª that contained the basis of span S that youobtained in part b).

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Answered: 1. Let S = {(1, 0, -1, -1),(1, -1, 1,…

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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Consider the basis B₁ = {(1,2), (3, -1)} of R2. Define T: R² = {(1, 0, 0), (1, 1, 0), (1, 1, 1)} of R³ and the basis B₂: R³ such that T((a, b)) = (a + 2b, - You may assume that T is linear. -a, -b). a. Find the matrix M(T) for T with respect to the standard bases in the domain and codomain. b. Find the matrix M(T) for T with respect to B₂ in the domain and B₁ in the codomain.
Consider the basis B = {({2, 1), (3, 1}} for V = R2 and let {x1, x2} be the dual basis of B. Then for an arbitrary pair (c, d) EV, the set {x1(c, d), x2(c,d)) is equal to: {(3d-c), (c-2d)} {(3d+ c), (c+ 2d)} {(2d-c), (c + 3d)} {(2d+c), (c + 3d)} {(3d+c), (2c - d)}
1. Determine whether the vectors (1,-1) and (1, 1)T are linearly inde- pendent or not. 2. Show that the vectors (1,0,0), (1, 1, 0) and (1,1,1) form a basis of R³. 3. Suppose that the vectors (1, 2, -1,0) and (1,3,2,0) span the sub- space U of R4. Determine whether the vector (1, 1, 1, 0) belongs to U or not.
just b)
3) Find a basis for R³ that includes a) the vector 0 B 2
Consider the basis B = {({2, 1), (3, 1}} for V = R2 and let {x1, x2} be the dual basis of B. Then for an arbitrary pair (c, d) EV, the set {x1(c, d), x2(c,d)) is equal to: {(3d-c), (c-2d)} {(3d+ c), (c+ 2d)} {(2d-c), (c + 3d)} {(2d+c), (c + 3d)} {(3d+c), (2c - d)}
1. Vectors v1 = (1,1,1,1,0), v2 = (-1,1,0,0,0), v3 = (1,1,-1,-1,0), v4 = (0,0,2,-2,0) and v5 = (0,0,0,0,2) form an orthogonal basis for a of R5 and X = (-3, -2,5,1,-4). Find the coordinates of X with respect to the given basis.
Q1) Let S (x-3, x² + 2x, x² + 1). Is S is span (P2(x)) or not? Q2) Let T= ((2,-2,4), (3,-5,4), (0,1,1)). Show that whether T is linearly independent or not? Q3) Let N = (vv) be a basis for vector space V Show that every vector in V can be written as uniquely linear combination of vectors of V.
Let W = Span(S), where | S = 1 -2 || -2 3 | -1 | Find a basis for W and calculate dim(W).
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)).
: Let S = {v1, v2}, where v = standard vectors when added to the set S produces a basis for R? (6,-5, 7,-1) and v2 = (-12, 20, -28, 4). Which of the following combination of (A) v3 = (1, 0, 0, 0), v4 = (0, 0, 1, 0) and vs = (0, 0, 0, 1) (B) v3 = (1, 0, 0, 0), v4 = (0, 1, 0, 0) and vs = (0, 0, 0, 1) (C) v3 = (1, 0, 0, 0) and v4 = (0, 0, 0, 1) (D) v3 = (1, 0, 0, 0), v4 = (0, 1, 0, 0) and vs = (0. 0, 1, 0) (E)v3 = (1, 0, 0, 0) and v4 = (0, 1, 0, 0) (F) v3 = (1, 0, 0, 0) and v4 = (0, 0, 1, 0) (G) v3 = (0, 1, 0, 0) and v4 = (0, 0, 0, 1) %3D %3D (H) v3 = (0, 1, 0, 0), v4 = (0, 0, 1, 0) and vs = (0. 0, 0, 1) %3D

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1. Let S = {(1, 0, -1, -1),(1, -1, 1, 2), (5, 2, -9, -11)} CR¹. a) Show that S' is linearly dependent over R. b) Determine a basis of Span (S) and dim (Span (S)) R c) Find a basis of R4 that contained the basis of span S that you obtained in part b).