Answer to Question #114348 in Linear Algebra for Peter

Question

2. Let A={b1,b2,b3} be a set of three-dimensional vectors in R3.
a. Prove that if the set A is linearly independent, then A is a basis of the vector space R3.

Answer 1

A set P is said to be a basis of a vector space if :

a) P is linearly independent set and

b) P is a spanning set of vector space.

Given that A is a linearly independent set.

To show that A is a basis, we need only prove that A is a spanning set of R³.

Let x∈R³ is an arbitrary vector.

To prove that A is a spanning set of R³, we will prove that there exist x₁,x₂,x₃

x₁b₁+x₂b₂+x₃b₃=x

This is equivalent to having a solution

x=[x_1, x₂, x₃]^-1 to the matrix equation

Ay=x,where A=[b₁,b₂,b₃] is the 3×3 matrix whose column vectors are b₁,b₂,b₃

Since the vector b₁, b₂, b₃ are linearly independent, the matrix A is nonsingular.

It follows that the equation Ay=x has the unique solution y=A^-1x

Hence x is a linear combination of the vectors in A.

This means that A is a spanning set of R³

Hence, A is a basis.