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)

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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