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field. For any A € M₂(K), let Let I₂ € M₂ (K) be the 2 × 2 identity matrix, where K is a V₁ = I2₁ V₂ = A₁ V3 = A² and regard them as elements of V = M₂(K) as a vector space. (a) Let K =R and A 61 (i) Show that v₁, U2, U3 are linearly dependent as elements of V. (ii) What is the dimension of the vector space (v1, U2, U3) spanned by V1, V2, V3? (b) Does there exist a choice of field K and of matrix A such that V₁, V2, V3 are linearly independent? You may wish to consider the characteristic polynomial PA(X).

field. For any A € M₂(K), let Let I₂ € M₂ (K) be the 2 × 2 identity matrix, where K is a V₁ = I2₁ V₂ = A₁ V3 = A² and regard them as elements of V = M₂(K) as a vector space. (a) Let K =R and A 61 (i) Show that v₁, U2, U3 are linearly dependent as elements of V. (ii) What is the dimension of the vector space (v1, U2, U3) spanned by V1, V2, V3? (b) Does there exist a choice of field K and of matrix A such that V₁, V2, V3 are linearly independent? You may wish to consider the characteristic polynomial PA(X).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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field. For any A € M₂(K), let Let I₂ € M₂ (K) be the 2 × 2 identity matrix, where K is a V₁ = I2, V₂ = A, V3 = A² and regard them as elements of V = M₂(K) as a vector space. 60 (a) Let K = R and A = (i) Show that v₁1, V2, V3 are linearly dependent as elements of V. (ii) What is the dimension of the vector space (v₁, V2, V3) spanned by V₁, V2, V3? (b) Does there exist a choice of field K and of matrix A such that V₁, V2, V3 are linearly independent? You may wish to consider the characteristic polynomial PA(x).
Transcribed Image Text:field. For any A € M₂(K), let Let I₂ € M₂ (K) be the 2 × 2 identity matrix, where K is a V₁ = I2, V₂ = A, V3 = A² and regard them as elements of V = M₂(K) as a vector space. 60 (a) Let K = R and A = (i) Show that v₁1, V2, V3 are linearly dependent as elements of V. (ii) What is the dimension of the vector space (v₁, V2, V3) spanned by V₁, V2, V3? (b) Does there exist a choice of field K and of matrix A such that V₁, V2, V3 are linearly independent? You may wish to consider the characteristic polynomial PA(x).
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