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Problem 3 Prove the following, ● Let f : R→ R be a linear map. Then show that f(x) = mx for some m Є R • Let f : Rn → R be a linear map. Then show that f(x1, x2,...,xn) = m1x1 m2x2 + for some mi Є R +mnxn Problem 4 4 4 Let M₁ = (28) and M₂ = (89) 99 6 6 " These both induce linear transformations M₁, M2 R² → R². Find a basis for its image and kernel. Find the rank, nullity and verify the rank-nullity theorem. Define h: R2 R2 R² (u, v) ↔ M₁u + M₂v Find a basis for its image and kernel. Find the rank, nullity and verify the rank-nullity theorem.

Problem 3 Prove the following, ● Let f : R→ R be a linear map. Then show that f(x) = mx for some m Є R • Let f : Rn → R be a linear map. Then show that f(x1, x2,...,xn) = m1x1 m2x2 + for some mi Є R +mnxn Problem 4 4 4 Let M₁ = (28) and M₂ = (89) 99 6 6 " These both induce linear transformations M₁, M2 R² → R². Find a basis for its image and kernel. Find the rank, nullity and verify the rank-nullity theorem. Define h: R2 R2 R² (u, v) ↔ M₁u + M₂v Find a basis for its image and kernel. Find the rank, nullity and verify the rank-nullity theorem.

Elementary Linear Algebra (MindTap Course List)
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 1CM
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Problem 3 Prove the following, ● Let f : R→ R be a linear map. Then show that f(x) = mx for some m Є R • Let f : Rn → R be a linear map. Then show that f(x1, x2,...,xn) = m1x1 m2x2 + for some mi Є R +mnxn
Transcribed Image Text:Problem 3 Prove the following, ● Let f : R→ R be a linear map. Then show that f(x) = mx for some m Є R • Let f : Rn → R be a linear map. Then show that f(x1, x2,...,xn) = m1x1 m2x2 + for some mi Є R +mnxn
Problem 4 4 4 Let M₁ = (28) and M₂ = (89) 99 6 6 " These both induce linear transformations M₁, M2 R² → R². Find a basis for its image and kernel. Find the rank, nullity and verify the rank-nullity theorem. Define h: R2 R2 R² (u, v) ↔ M₁u + M₂v Find a basis for its image and kernel. Find the rank, nullity and verify the rank-nullity theorem.
Transcribed Image Text:Problem 4 4 4 Let M₁ = (28) and M₂ = (89) 99 6 6 " These both induce linear transformations M₁, M2 R² → R². Find a basis for its image and kernel. Find the rank, nullity and verify the rank-nullity theorem. Define h: R2 R2 R² (u, v) ↔ M₁u + M₂v Find a basis for its image and kernel. Find the rank, nullity and verify the rank-nullity theorem.
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