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(a) State the formal mathematical definition of linear independence for a set of N vec- tors, and explain with a sketch what requirements this would place on two vectors in two dimensional space.

(a) State the formal mathematical definition of linear independence for a set of N vec- tors, and explain with a sketch what requirements this would place on two vectors in two dimensional space.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 3AEXP
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(a) State the formal mathematical definition of linear independence for a set of N vec- tors, and explain with a sketch what requirements this would place on two vectors in two dimensional space. Consider the equation: Ag = b, where A is a knownn x n matrix, b is a known vector and a is a vector containing an unknown set of solutions. (b) State clearly one method to compute the determinant of the n x n matrix A (c) What conditions must be satisified by matrix A such that equation Ag = b has exactly one solution for the unknown vector z? (d) Represent the following simultaneous equations in the form: Ar = b , 3.r1 + 2x2 + ag = 13 , a1 – 3x2 + 2rg = -5, 271 + x2 + 3xg = 10 . (e) Stating clearly Cramer's rule, find the matrix inverse, and use it to determine a unique solution to the equations in part (d). (f) We have a new set of simultaneous equations (given below). Find the condition(s) on a such that the simultaneous equations: xi + ar2 = 1, x1 - *2 + 3x3 = -1, 2.x1 – 2x2 + ar3 = -2, have exactly one solution, and determine the vector solution, z in terms of a. (g) Consider the following matrices: 0 -i V3 -V2 -V3 M1 = M2 V8 i -i 1 -1 i 0 2 State whether these matrices are: (i) symmetric, (ii) orthogonal, (iii) Hermitian, (iv) unitary. Justify your answers.
Transcribed Image Text:(a) State the formal mathematical definition of linear independence for a set of N vec- tors, and explain with a sketch what requirements this would place on two vectors in two dimensional space. Consider the equation: Ag = b, where A is a knownn x n matrix, b is a known vector and a is a vector containing an unknown set of solutions. (b) State clearly one method to compute the determinant of the n x n matrix A (c) What conditions must be satisified by matrix A such that equation Ag = b has exactly one solution for the unknown vector z? (d) Represent the following simultaneous equations in the form: Ar = b , 3.r1 + 2x2 + ag = 13 , a1 – 3x2 + 2rg = -5, 271 + x2 + 3xg = 10 . (e) Stating clearly Cramer's rule, find the matrix inverse, and use it to determine a unique solution to the equations in part (d). (f) We have a new set of simultaneous equations (given below). Find the condition(s) on a such that the simultaneous equations: xi + ar2 = 1, x1 - *2 + 3x3 = -1, 2.x1 – 2x2 + ar3 = -2, have exactly one solution, and determine the vector solution, z in terms of a. (g) Consider the following matrices: 0 -i V3 -V2 -V3 M1 = M2 V8 i -i 1 -1 i 0 2 State whether these matrices are: (i) symmetric, (ii) orthogonal, (iii) Hermitian, (iv) unitary. Justify your answers.
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