For the subspace below, (a) find a basis for the subspace, and (b) state the dimension.{(а, b, с, d): a — 5b + 6с %3D 0}(a) Find a basis for the subspace.A basis for the subspace is { }.(Use a comma to separate matrices as needed.)(b) State the dimension.The dimension is

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Answered: For the subspace below, (a) find a…

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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Let A = {a₁,32,33} and B = {b₁,b₂,b3} be bases for a vector space V, and suppose a₁ = 5b₁ b₂, 1 a₂ = -b₁ +4b₂ + b3, a3 = b₂ − 3b3- a. Find the change-of-coordinates matrix from A to B. b. Find [x]g for x = 4a₁ +5a₂ + a3.
Consider the following matrix: 2 -2 4 6 6 A = -1 1 3 2 -3 -1 1 -3 -3 1 Give a basis for each of col(A) and null(A). Number of Vectors: 1 {} Basis for col(A)||0 Number of Vectors: 1 {8} 0 Basis for null(A) {
I1 Find a basis for the subspace of R³ consisting of all vectors such that 8₁-712+673 = 0. Hint: Notice that this single equation counts as a system of linear equations; find and describe the solutions. Answer:
Let W be the subspace of all diagonal matrices in M2.2. Find a basis for W. Then give the dimension of W. If you need to enter a matrix as part of your answer, write each row as a vector. For example, write the matrix ab c d as {(a, b), (c, d)}
The standard basis & (e₁,e₂) and two custom bases B (b₁,b₂} and C (c₁, c₂} for R² are shown in the figures below. Standard basis S= (₁, ₂) 4 a. Find the change of basis matrix from custom B-coordinates to standard S-coordinates. [id] = id= -2 b Find the change of basis matrix from custom C-coordinates to standard S-coordinates [id] = -1 -1 -1 c. Find the change of basis matrix from B-coordinates to C-coordinates -3 -1/12 1/4 d. Find the change of basis matrix from C-coordinates to B-coordinates. 3 2 1 -1 -2 -9 b2 -3-2-1 82 el bl 1 2 3 b1 X Custom basis B = {b₁,b₂) |id|s [id Standard basis S= (0₁,0₂} a 3 2 1 -1 -2 -9 c2 -3 -2 -1 G2 62 217 el [id] ↑ cl 12 2 3 61 H x Custom basis C (C₁, C₂)
Let v1, V2 and v3 be three linearly independent vectors in R3. (a) Find the rank of the matrix A = [(v1 – v2) (v2 – v3) (V3 – V1)]. (b) Find the rank of the matrix B = [(v1 + v2) (V2 + V3) (V3 + V1)].
For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A. 0 --[142-9] A = -5 A nonzero vector in Nul A is (Type an integer or decimal for each matrix element.) A nonzero vector in Col A is. (Type an integer or decimal for each matrix element.)
4. Describe all matrices row equivalent to the matrix: 3 -2 -[:] [] 0 2 3 (a) List five vectors in Span{V₁, V₂}. For each vector, give the corresponding weights. (b) Give a geometric description of Span{V1, V2}. 5. Consider the vectors: V₁ = and v₂ = 1
For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A. A = 2 -2 4 10 - 16 16 2 -26 40 4 -8-2

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For the subspace below, (a) find a basis for the subspace, and (b) state the dimension. {(a, b, c, d): a - 5b + 6c = 0} .. (a) Find a basis for the subspace. A basis for the subspace is (Use a comma to separate matrices as needed.) (b) State the dimension. The dimension is.