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Problem 2 Let U, V, W, Z be vector spaces. Let f : U → W, g : V → Z be two linear maps. Define the map, h: Ux VW × Z (v, w) (f(v), g(w)) where addition and scalar multiplication on a product U × V is defined componentwise (see previous tutorial for a refresher). Prove that, • h is a linear map • ker(h) = ker(ƒ) × ker(g) • Im(h) = Im(ƒ) × Im(g). Problem 1 Find a basis for the kernel and the image of the following maps. Also find the dimension of the kernel and image. (a) Matrix A (defined below) be a linear map A: R4 → R³ : (b) Let G R³ R³ be the map 1 2 3 1 A: = 1 3 5 -2 3 8 13 -3 - G(x, y, z) = (x + 2y − z, y + z, x + y − 2z) -

Problem 2 Let U, V, W, Z be vector spaces. Let f : U → W, g : V → Z be two linear maps. Define the map, h: Ux VW × Z (v, w) (f(v), g(w)) where addition and scalar multiplication on a product U × V is defined componentwise (see previous tutorial for a refresher). Prove that, • h is a linear map • ker(h) = ker(ƒ) × ker(g) • Im(h) = Im(ƒ) × Im(g). Problem 1 Find a basis for the kernel and the image of the following maps. Also find the dimension of the kernel and image. (a) Matrix A (defined below) be a linear map A: R4 → R³ : (b) Let G R³ R³ be the map 1 2 3 1 A: = 1 3 5 -2 3 8 13 -3 - G(x, y, z) = (x + 2y − z, y + z, x + y − 2z) -

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.6: Rank Of A Matrix And Systems Of Linear Equations
Problem 68E: Show that the three points (x1,y1)(x2,y2) and (x3,y3) in the a plane are collinear if and only if...
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Problem 2 Let U, V, W, Z be vector spaces. Let f : U → W, g : V → Z be two linear maps. Define the map, h: Ux VW × Z (v, w) (f(v), g(w)) where addition and scalar multiplication on a product U × V is defined componentwise (see previous tutorial for a refresher). Prove that, • h is a linear map • ker(h) = ker(ƒ) × ker(g) • Im(h) = Im(ƒ) × Im(g).
Transcribed Image Text:Problem 2 Let U, V, W, Z be vector spaces. Let f : U → W, g : V → Z be two linear maps. Define the map, h: Ux VW × Z (v, w) (f(v), g(w)) where addition and scalar multiplication on a product U × V is defined componentwise (see previous tutorial for a refresher). Prove that, • h is a linear map • ker(h) = ker(ƒ) × ker(g) • Im(h) = Im(ƒ) × Im(g).
Problem 1 Find a basis for the kernel and the image of the following maps. Also find the dimension of the kernel and image. (a) Matrix A (defined below) be a linear map A: R4 → R³ : (b) Let G R³ R³ be the map 1 2 3 1 A: = 1 3 5 -2 3 8 13 -3 - G(x, y, z) = (x + 2y − z, y + z, x + y − 2z) -
Transcribed Image Text:Problem 1 Find a basis for the kernel and the image of the following maps. Also find the dimension of the kernel and image. (a) Matrix A (defined below) be a linear map A: R4 → R³ : (b) Let G R³ R³ be the map 1 2 3 1 A: = 1 3 5 -2 3 8 13 -3 - G(x, y, z) = (x + 2y − z, y + z, x + y − 2z) -
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