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Linear Independence and Invertible Grammatian Matrices, Assignments of Linear Algebra

University of California - BerkeleyLinear Algebra

A homework assignment from math 110, section 7, fall 2003, focusing on the relationship between linear independence of vectors in an inner product space and the invertibility of their grammatian matrices. The assignment includes exercises to prove this theorem, with one problem asking to prove the equivalence of linear independence and invertible grammatian matrices.

Typology: Assignments

Pre 2010

Uploaded on 10/01/2009

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Math 110, Section 7 Fall 2003
Sarason
HOMEWORK ASSIGNMENT 8
Due in class on Friday, October 31.
Pages 124–125, Exercises 21, 24, 26, 28, 29, 30, 31, plus Exercise G below.
G. Let v1,v
2,...,v
nbe vectors in an inner product space. The Grammian G(v1,...,v
n)of
v1,...,v
nis the n×nmatrix whose (j, k)th entry is vj,v
k. Prove that v1,...,v
nare linearly
independent if and only if G(v1,...,v
n)isinvertible.

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Math 110, Section 7 Fall 2003 Sarason

HOMEWORK ASSIGNMENT 8

Due in class on Friday, October 31.

Pages 124–125, Exercises 21, 24, 26, 28, 29, 30, 31, plus Exercise G below.

G. Let v 1 , v 2 ,... , vn be vectors in an inner product space. The Grammian G(v 1 ,... , vn) of v 1 ,... , vn is the n × n matrix whose (j, k)th^ entry is 〈vj , vk〉. Prove that v 1 ,... , vn are linearly independent if and only if G(v 1 ,... , vn) is invertible.