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(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. Justify your answers. A = A = 29081 28602 52747 215 620 109 518 936

(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. Justify your answers. A = A = 29081 28602 52747 215 620 109 518 936

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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Needed only C part correctly Please solve correctly by hand solution needed Please only Part C
Question 1 (a) Find the rank of the following matrix for all values of a a -1 2 1 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 = 2908 1 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 - x² in this basis. Show your working.
Transcribed Image Text:Question 1 (a) Find the rank of the following matrix for all values of a a -1 2 1 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 = 2908 1 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 - x² in this basis. Show your working.
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