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Let V and W be vector spaces over R and let L:V → W be a linear mạp. Recall the kernel of L is defined as Ker(L) = {x E V | Lx = 0} and the image of L is defined as Im(L) = {y € W| y = L(x), 3x E V}. Prove Ker(L) is a subspace of V and Im(L) is a subspace of W. What is the relationship between the dimensions of these subspaces, when V is finite dimensional?

Let V and W be vector spaces over R and let L:V → W be a linear mạp. Recall the kernel of L is defined as Ker(L) = {x E V | Lx = 0} and the image of L is defined as Im(L) = {y € W| y = L(x), 3x E V}. Prove Ker(L) is a subspace of V and Im(L) is a subspace of W. What is the relationship between the dimensions of these subspaces, when V is finite dimensional?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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Let V and W be vector spaces over R and let L:V → W be a linear mạp. Recall the kernel of L is defined as Ker(L) = {x E V | Lx = 0} and the image of L is defined as Im(L) = {y € W| y = L(x), 3x E V}. Prove Ker(L) is a subspace of V and Im(L) is a subspace of W. What is the relationship between the dimensions of these subspaces, when V is finite dimensional?
Transcribed Image Text:Let V and W be vector spaces over R and let L:V → W be a linear mạp. Recall the kernel of L is defined as Ker(L) = {x E V | Lx = 0} and the image of L is defined as Im(L) = {y € W| y = L(x), 3x E V}. Prove Ker(L) is a subspace of V and Im(L) is a subspace of W. What is the relationship between the dimensions of these subspaces, when V is finite dimensional?
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