Given x1 = (1,1,1)T and x2 = (3,−1,4)T: (a) Do x1 and x2 span R3 ? Explain. (b) Let x3 be a third vector in R3and set X = (x1 x2 x3). What condition(s) would X have to satisfy in order for x1, x2, and x3 to form a basis for R3?

Given x1 = (1,1,1)T and x2 = (3,−1,4)T: (a) Do x1 and x2 span R3 ? Explain. (b) Let x3 be a third vector in R3and set X = (x1 x2 x3). What condition(s) would X have to satisfy in order for x1, x2, and x3 to form a basis for R3?

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.1: Orthogonality In Rn

Problem 7EQ

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Question

Given x₁= (1,1,1)^Tand x₂ = (3,−1,4)^T:

(a) Do x₁ and x₂ span R³ ? Explain.

(b) Let x₃ be a third vector in R³and set X = (x₁x₂x₃). What condition(s) would X have to satisfy in order for x₁, x₂, and x₃ to form a basis for R³?

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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