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A. Orthogonal projection onto the y-axis in R². Consider the linear transformation that projects all points in the plane horizontally onto the y-axis. The point (x, y) maps to point (0, y). We can define the transformation T: R² R² by T(x) = Ax for some 2 x 2 matrix A. 0 0 0 0 0 Remember that if Ker(T) is trivial, then it has no basis. Select vectors that form a basis for Ker(T), if such a basis exists. There is no basis for the kernel of T. [8] ( H H G] H Pre-image (before transformation) [2] [7] [2] 0 K(0,3) M(0,2) (a,b) 2 L(2,2.5) 3 Image (after transformation) 24 4 O 4 K'(0,3) L'(0,2.5) M'(0,2) J'(0,0) 1 2 3

A. Orthogonal projection onto the y-axis in R². Consider the linear transformation that projects all points in the plane horizontally onto the y-axis. The point (x, y) maps to point (0, y). We can define the transformation T: R² R² by T(x) = Ax for some 2 x 2 matrix A. 0 0 0 0 0 Remember that if Ker(T) is trivial, then it has no basis. Select vectors that form a basis for Ker(T), if such a basis exists. There is no basis for the kernel of T. [8] ( H H G] H Pre-image (before transformation) [2] [7] [2] 0 K(0,3) M(0,2) (a,b) 2 L(2,2.5) 3 Image (after transformation) 24 4 O 4 K'(0,3) L'(0,2.5) M'(0,2) J'(0,0) 1 2 3

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 22EQ
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A. Orthogonal projection onto the y-axis in R². Consider the linear transformation that projects all points in the plane horizontally onto the y-axis. The point (x, y) maps to point (0, y). We can define the transformation T: R² R² by T(x) = Ax for some 2 x 2 matrix A. U 0 0 U Remember that if Ker(T) is trivial, then it has no basis. Select vectors that form a basis for Ker(T), if such a basis exists. U 0 U 0 U There is no basis for the kernel of T. [] H GJ [) H [1] Pre-image Image (before transformation) (after transformation) [2¹] 0 K(0,3) M(0, 2) J(0,0) L(2,2.5) -1 3 24 1 0 - K'(0,3) L'(0,2.5) M'(0, 2) J'(0,0) 2 3
Transcribed Image Text:A. Orthogonal projection onto the y-axis in R². Consider the linear transformation that projects all points in the plane horizontally onto the y-axis. The point (x, y) maps to point (0, y). We can define the transformation T: R² R² by T(x) = Ax for some 2 x 2 matrix A. U 0 0 U Remember that if Ker(T) is trivial, then it has no basis. Select vectors that form a basis for Ker(T), if such a basis exists. U 0 U 0 U There is no basis for the kernel of T. [] H GJ [) H [1] Pre-image Image (before transformation) (after transformation) [2¹] 0 K(0,3) M(0, 2) J(0,0) L(2,2.5) -1 3 24 1 0 - K'(0,3) L'(0,2.5) M'(0, 2) J'(0,0) 2 3
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