x + 2y + 2wz+ wLet W =e, y, z, w E R } be a- 3. – 6y + 4z - 2wsubspace of R4. What is the dimension of W?dim(W) = 3dim(W) = 4dim(W) = 1dim(W) = 2

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Answered: a+ 2y + 2w z+ w Let W = x, y, z, w E R…

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 W be the subspace spanned by u₁ and u₂, and write y as the sum of a vector in W and a vector orthogonal to W. -0- y = -2 3,4₁ 5 The sum is y = y + z, where y= (Simplify your answers.) -3 -1 is in W and Z= is orthogonal to W.
Let W be the set of all vectors in R² With second component equal to twice the first, that all vectors of the form (9,20) (a) Use the Test for a Subspace to show that W is a subspace of R². (6) Using the x-y plane to represent R² a sketch of W. make
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U₁ = 4 #0-0 -2,1₂ = and и3 = -32 Note that u₁, ₂ and 3 are orthogonal. V = Given v = 4 -0₁ 2 find the coordinates for v in the subspace Wspanned by 4 4 4 U₁ + -8 -8 U₂ +1 U3|
Let W- ((x. y, 2))x- 2y = 32) Vector addition and scalar multiplication are defined by (x. y.z) + (u, v, w) - (x + u. y+w-w) k (x, y, a) - (kx, ky, k'2) Determine whether W is a subspace of R.if so, prove it. If not, explain why it is not a subspace. 2. (a) By inspection (withaut any calculation) determine whether the following set is independent (b) Determine whether the vectors u= (1,-3,7), v- (3,-1-1), and w (2,4,-5) in R form a basis set for R. Determine whether the polynomials u=2+2x+ 4zx,1+, and w 3+ 2x + 5x span P, (the set of polynomials of degree 2 or less). If not find the space spanned by the given vectors. 3. Let S- (v, vz, vy, v4), where 4. and v Find a basis for the space spanned by the space span(5) which is a subset of S. 5. -1 1 The eigenvalues of the matrix A1 2. 1 are -1-2, and 3. Also, the eigenvector corresponds to the eigenvalues are respectively and (a) Find the matrix P that diagonalizes A (b) Find the diagonal matrix D (c) Hence compute A0
V = {(x, y) E R² | x = y} u {(x, y) E R² | x = = -2y}. Complete the following statements to determine if V is a subspace of R. (a) V is ? If it is non-empty, give two different examples of vectors in V. If it is empty, then leave the following spaces blank. example 1: aа — example 2: b = Note: Normally, only one example is required to show V is not empty in a proof. (b) V is ? v under vector addition. If it is not closed, enter two vectors a, b E V below, whose sum is not in V. If it is closed, then leave the following spaces blank. a = b = (c) V is ? under scalar multiplication. If it is not closed, enter a scalar k and a vector c E V below, whose product is not in V. If it is closed, then leave the following spaces blank. (OO) k = and c = (d) V ? of R?.
9 Find a basis and the dimension for the vector subspace W = ((x,y,z) e R³/ 2x-y+2=0 -8x+4y-4z=0} -6x+3y-3z0
Question (3): Find the basis and dimension of the subspace W: 20 а — b a a) W =- :a, b, c eR }e Rª. b – 3c La + 2b b) W = { :x ER} € M2x2-
Let fi = 1+2x + 3x², f2 = 2 + 3x + 4x². B = (f1, f2) is a basis for the subspace V = {ao +a1x+azx²[ao – 2a1 +a2 = 0} of R2[r]. Let g = 1+5x + 9x². If [g]g = (,) then s = a) -7 b) -5 c) 5 d) 7 e) g is not in V.

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a+ 2y + 2w z+ w Let W = x, y, z, w E R be a –32 – 6y + 4z – 2w z- w subspace of R. What is the dimension of W? O dim(W) = 3 dim(W) = 4 dim(W) = 1 dim(W) = 2