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1. Let ƒ : R³ → R5 be the function given by f(x₁, x2, X3) = (x₁ + x2 − X3, 2x1 + 2x2 − X3, 3X1 + 3x2 - X3, X1 X2 X3, X3) - - - a.) Find the matrix A for which f(x) = Ax. b.) Find all vectors (1, 2, 3) in R³ for which f(x1, x2, 3) = (1, 1, 1, 1, -1).

1. Let ƒ : R³ → R5 be the function given by f(x₁, x2, X3) = (x₁ + x2 − X3, 2x1 + 2x2 − X3, 3X1 + 3x2 - X3, X1 X2 X3, X3) - - - a.) Find the matrix A for which f(x) = Ax. b.) Find all vectors (1, 2, 3) in R³ for which f(x1, x2, 3) = (1, 1, 1, 1, -1).

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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1. Let f: R³ R5 be the function given by f(x1, x2, 3) = (x₁ + x₂ - x3,2x1 + 2x2 - x3, 3x1 + 3x2x3, x1 - x2 - X3, X3) a.) Find the matrix A for which f(x) = Ax. b.) Find all vectors (x1, x2, 3) in R³ for which f(x1, x2, 3) = (1, 1, 1, 1, -1). c.) State a basis for the column space of A (i.e., the range of f). What is the dimension of the range of f? d.) State a basis for the null space of A (i.e., the kernel of f). What is the dimension of the kernel of f?
Transcribed Image Text:1. Let f: R³ R5 be the function given by f(x1, x2, 3) = (x₁ + x₂ - x3,2x1 + 2x2 - x3, 3x1 + 3x2x3, x1 - x2 - X3, X3) a.) Find the matrix A for which f(x) = Ax. b.) Find all vectors (x1, x2, 3) in R³ for which f(x1, x2, 3) = (1, 1, 1, 1, -1). c.) State a basis for the column space of A (i.e., the range of f). What is the dimension of the range of f? d.) State a basis for the null space of A (i.e., the kernel of f). What is the dimension of the kernel of f?
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