3. Each of J, K, L, M and N is a linear transformation from R' to R?. These functions are given as follows: J(11, 12) = (5x, – 3z2, –10r, + 6z2), K(1, T2) = (-22, T), L(I1, 12) = (2, 1), M(x1, 12) = (3r, + 12, I1 + 2ra), N(11, 12) = (-11, 22). %3D (a) In each case, write down the matrix of the transformation and compute its deter- minant. (b) Sketch a picture indicating what happens to the standard basis {(1,0). (0. 1)} un- der each function. (c) One of these functions is not injective. Which is it? (d) One of these functions is an anti-clockwise rotation of the plane. Which is it? (e) One of these functions is a reflection over the vertical axis. Which is it?
3. Each of J, K, L, M and N is a linear transformation from R' to R?. These functions are given as follows: J(11, 12) = (5x, – 3z2, –10r, + 6z2), K(1, T2) = (-22, T), L(I1, 12) = (2, 1), M(x1, 12) = (3r, + 12, I1 + 2ra), N(11, 12) = (-11, 22). %3D (a) In each case, write down the matrix of the transformation and compute its deter- minant. (b) Sketch a picture indicating what happens to the standard basis {(1,0). (0. 1)} un- der each function. (c) One of these functions is not injective. Which is it? (d) One of these functions is an anti-clockwise rotation of the plane. Which is it? (e) One of these functions is a reflection over the vertical axis. Which is it?
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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part d
Transcribed Image Text:3. Each of J, K, L, M and N is a linear transformation from R to R', These functions
are given as follows:
J(11, 12) = (5x1 – 3z2, –10x1 + 672),
K(x1, I2) = (-12, 1,),
L(I1, 72) = (2, 1),
M(I1,12) = (311 + I2, I1 + 2r2),
N(r,, 12) = (-1,12).
(a) In each case, write down the matrix of the transformation and compute its deter-
minant.
(b) Sketch a picture indicating what happens to the standard basis {(1,0), (0, 1)} un-
der each function.
(c) One of these functions is not injective. Which is it?
(d) One of these functions is an anti-clockwise rotation of the plane. Which is it?
(e) One of these functions is a reflection over the vertical axis. Which is it?
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