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Let be a basis for a subspace of R2, 12 the Gram-Schmidt process to find an orthogonal pasis under the Frobenjus inner product. 1.38 0.46 Orthogonal basis Ex: 5 b= Ex 5 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
U2
be a basis for a subspace of IR2X2
12
the Gram-Schmidt process to find an orthogonal pasis under the Frobenjus inner product.
1.38
Orthogonal basis
Ex: 5
b= Ex 5
4.
Check

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(2, 4, 0), the Gram-Schmidt process Given vị = (-1, 2, –1), v2 = (1, 7,1) and v3 allows us to construct an orthonormal basis {u1, u2, U3 } of R³ such that u¡ is a multiple of V1, u2 is a linear combination of vj and v2, and uz is a linear combination of v1, V2 and v3. Then u2 is equal to: O (1,1, 1) O (1,2, 0) (1:2'0뚜 이 O (0, 1, 0) o (-1, 2, –1) ㅇ 늘(1,0, 1)
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*1. Let W be the subspace of R5 spanned by the vectors B Find a basis for W. Give the dimension of W. -3 b₁ = 2 *2. Consider B = {b₁,b₂}, a basis for R² where - (²). b₂ = (3) 0 (a) Find the change of basis matrix that converts vectors given with respect to B into vectors given with respect to the standard basis. Give your matrix explicitly. (b) Find the change of basis matrix that converts vectors given with respect to the standard basis into vectors given with respect to B. Give your matrix explicitly. 3. True or False? (a) There is a 5 x 3 matrix whose column space is 1-dimensional. (b) If A is an m x n matrix, then rank(A) + nullity(A) = m. (c) If A is an n x n matrix and the columns of A span R", then A is invertible. (d) If B is a 2 x 2 matrix and B² = I₂, then B = I2. (e) The kernel of the function T: R² R² given by T + (²) - (² + 1) ₁ = is {0}. (f) There is a matrix that has a 3-dimensional column space and a 2-dimensional row space.
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