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Question 1 Find an orthonormal basis for the solution space of the given homogeneous system of linear equations. w+x-y-z=0 2w+x-2y-2z = 0.

Question 1 Find an orthonormal basis for the solution space of the given homogeneous system of linear equations. w+x-y-z=0 2w+x-2y-2z = 0.

Advanced Engineering Mathematics
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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Question 1 Find an orthonormal basis for the solution space of the given homogeneous system of linear equations. w+x-y-z=0 2w+x-2y-2z = 0. Question 2 Is (u, v) = au₁v₁ + b₁V₂ + bu₂v₁ + c₂V₂, where u = (₁, ₂) and v = (₁, ₂) are vectors in R², an inner product in the vector space R²? Show all details of your work. Question 3 -1 The linear transformation T is given by T(x) = Ax, where A = 5 6 nullity, range, and rank of the linear transformation. 4 Question 4 Find the least squares solution of the linear system Ax = b given by x-y=4 3x + 2y = 1 -2x + 4y = 3. 3 -4. Find the kernel, 2 Question 5 Let T: R² → R² be the reflection through the x-axis. a) Find the standard matrix of T. b) Show that I is an isomorphism and find a formula for T-¹. c) Consider the basis {(1, 2), (1,0)) of R². Find the matrix of T relative to this basis.
Transcribed Image Text:Question 1 Find an orthonormal basis for the solution space of the given homogeneous system of linear equations. w+x-y-z=0 2w+x-2y-2z = 0. Question 2 Is (u, v) = au₁v₁ + b₁V₂ + bu₂v₁ + c₂V₂, where u = (₁, ₂) and v = (₁, ₂) are vectors in R², an inner product in the vector space R²? Show all details of your work. Question 3 -1 The linear transformation T is given by T(x) = Ax, where A = 5 6 nullity, range, and rank of the linear transformation. 4 Question 4 Find the least squares solution of the linear system Ax = b given by x-y=4 3x + 2y = 1 -2x + 4y = 3. 3 -4. Find the kernel, 2 Question 5 Let T: R² → R² be the reflection through the x-axis. a) Find the standard matrix of T. b) Show that I is an isomorphism and find a formula for T-¹. c) Consider the basis {(1, 2), (1,0)) of R². Find the matrix of T relative to this basis.
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