Question

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.

b. Prove that if the set A spans R3, then A is a basis of R3.

Answer 1

Given that ,

$A=\{ b_1,b_2,b_3 \}$ be a set of three dimensional vector in $\R^3$ .

Since , $\R^3$ is a Vector space over $\R$ then $dim(\R^3)=3.$

Again we known that , If $V$ is a Vector Space of finite dimension n .Then 1) Any linearly independent set $S=\{ u_1,u_2,............,u_n \}$

with n elements is a basis of V.

2) Any spanning set $T=\{ v_1,v_2,......,v_n\}$ of $V$ with n elements is a basis of $V$ .

Since cardinality of $A$ is 3 ,Hence by (1) , $A$ is a basis of $\R^3$ and if the set $A$ spanning $\R^3$ by (2) , $A$ is a basis of $\R^3$ .