Problem 16. Let T : VW be a linear transformation between vector spaces of the same dimension. Prove that if T is injective, then it is invertible. Prove, that if T is surjective, then it is invertible. Problem 2. Consider the trace map T : M2×2 → R defined by Prove that T is a linear transformation. T = a + d. A very important fact about linear transformations is that they are completely determined by what they do to a basis. More formally: given vectors U₁,..., Un in a vector space V and vectors w₁,..., W₂ in a vector space W, there exists a unique linear transformation T: VW such that T(vi) = Wi. Problem 3. Consider problem 5.9 in Schaum's. Take a basis {1, 2} of R² of your choice and find the value of the linear transformation on the vectors in the basis, i.e. find F(1), F(2). Express F of F(1) and F(№2). as a combination
Problem 16. Let T : VW be a linear transformation between vector spaces of the same dimension. Prove that if T is injective, then it is invertible. Prove, that if T is surjective, then it is invertible. Problem 2. Consider the trace map T : M2×2 → R defined by Prove that T is a linear transformation. T = a + d. A very important fact about linear transformations is that they are completely determined by what they do to a basis. More formally: given vectors U₁,..., Un in a vector space V and vectors w₁,..., W₂ in a vector space W, there exists a unique linear transformation T: VW such that T(vi) = Wi. Problem 3. Consider problem 5.9 in Schaum's. Take a basis {1, 2} of R² of your choice and find the value of the linear transformation on the vectors in the basis, i.e. find F(1), F(2). Express F of F(1) and F(№2). as a combination
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 5CM: Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
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Transcribed Image Text:Problem 16. Let T : VW be a linear transformation between vector spaces of the same dimension. Prove
that if T is injective, then it is invertible. Prove, that if T is surjective, then it is invertible.
Transcribed Image Text:Problem 2. Consider the trace map T : M2×2 → R defined by
Prove that T is a linear transformation.
T
= a + d.
A very important fact about linear transformations is that they are completely determined by what they do to
a basis. More formally: given vectors U₁,..., Un in a vector space V and vectors w₁,..., W₂ in a vector space
W, there exists a unique linear transformation T: VW such that T(vi)
= Wi.
Problem 3. Consider problem 5.9 in Schaum's. Take a basis {1, 2} of R² of your choice and find the value of
the linear transformation on the vectors in the basis, i.e. find F(1), F(2). Express F
of F(1) and F(№2).
as a combination
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