Answer to Question #114572 in Linear Algebra for Peter

Question #114572
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.
1
Expert's answer
2020-05-11T12:17:03-0400

Given that ,

A={b1,b2,b3}A=\{ b_1,b_2,b_3 \} be a set of three dimensional vector in R3\R^3 .

Since , R3\R^3 is a Vector space over R\R then dim(R3)=3.dim(\R^3)=3.

Again we known that , If VV is a Vector Space of finite dimension n .Then 1) Any linearly independent set S={u1,u2,............,un}S=\{ u_1,u_2,............,u_n \}

with n elements is a basis of V.

2) Any spanning set T={v1,v2,......,vn}T=\{ v_1,v_2,......,v_n\} of VV with n elements is a basis of VV .

Since cardinality of AA is 3 ,Hence by (1) , AA is a basis of R3\R^3 and if the set AA spanning R3\R^3 by (2) , AA is a basis of R3\R^3 .


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