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A transformation f: R³ → R² is defined by f (x1, x2, X3) = (x₁ − x₂ + 2x3, 2x1 − X2 −X3). i. Show that f is a linear transformation. ii. Write down the standard matrix of f with respect to the standard bases of R³ and R². iii. Show that f is not a one-to-one transformation. iv. Find the kernel of f.

A transformation f: R³ → R² is defined by f (x1, x2, X3) = (x₁ − x₂ + 2x3, 2x1 − X2 −X3). i. Show that f is a linear transformation. ii. Write down the standard matrix of f with respect to the standard bases of R³ and R². iii. Show that f is not a one-to-one transformation. iv. Find the kernel of f.

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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A transformation f: R³ → R² is defined by f (x1, x2, X3) = (x₁ − x₂ + 2x3,2x1 − X2 − X3). i. Show that f is a linear transformation. ii. Write down the standard matrix of f with respect to the standard bases of R³ and R². iii. Show that f is not a one-to-one transformation. iv. Find the kernel of f.
Transcribed Image Text:A transformation f: R³ → R² is defined by f (x1, x2, X3) = (x₁ − x₂ + 2x3,2x1 − X2 − X3). i. Show that f is a linear transformation. ii. Write down the standard matrix of f with respect to the standard bases of R³ and R². iii. Show that f is not a one-to-one transformation. iv. Find the kernel of f.
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