(a) Find the rank of the following matrix for all values of a 1 a -1 2 2-1 a 5 1 10 -61 Show your working. (b) Consider the complex vector space C² (with complex scalars). Find the real number a, if it exists, such that the following two vectors V₁ and V₂ in C² are linearly dependent: V₁ = (1 + i, 3-i), V₂ = (-2-i, a-3i). (c) For each of the following matrices A, decide whether there exist distinct non-zero vectors x₁ and x₂ of appropriate dimensions such that T₁(x₁) = TA(X2). i. ii. A = Justify your answers. (d) Show that the polynomials . A = 29081 28602 52747 215 620 109 518 936 P₁ = 3+2x+x², P2=2-x+3x², P3= −2+x-2x² form a basis of the vector space P₂ of polynomials of degree at most 2 and find the coordinates of 1+ 2x - 2² in this basis. Show your working.

(a) Find the rank of the following matrix for all values of a 1 a -1 2 2-1 a 5 1 10 -61 Show your working. (b) Consider the complex vector space C² (with complex scalars). Find the real number a, if it exists, such that the following two vectors V₁ and V₂ in C² are linearly dependent: V₁ = (1 + i, 3-i), V₂ = (-2-i, a-3i). (c) For each of the following matrices A, decide whether there exist distinct non-zero vectors x₁ and x₂ of appropriate dimensions such that T₁(x₁) = TA(X2). i. ii. A = Justify your answers. (d) Show that the polynomials . A = 29081 28602 52747 215 620 109 518 936 P₁ = 3+2x+x², P2=2-x+3x², P3= −2+x-2x² form a basis of the vector space P₂ of polynomials of degree at most 2 and find the coordinates of 1+ 2x - 2² in this basis. Show your working.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 29E

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