Answer to Question #115178 in Linear Algebra for Peter

Question

1. Verify if the vectors (3, 4, 5), (-3, 0, 5), (4, 4, 4), (3, 4, 0) are linearly independent.
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.
b. Prove that if the set A spans R3, then A is a basis of R3.

Answer 1

1)Verify if the vectors (3, 4, 5), (-3, 0, 5), (4, 4, 4), (3, 4, 0) are linearly independent.

Solution:

The vectors are linearly dependent because the dimension of the vectors is less than the number of vectors.

2)a. Prove that if the set A is linearly independent, then A is a basis of the vector space R3.

 

Any finite-dimensional vector space has a basis.

By the condition, there is a finite generating system of vectors of a given finite-dimensional vector space A.

We can assume that all vectors of this system are non-zero, because otherwise, we can remove them from this system and the remaining system of vectors will be a finite generating system.

Note immediately that if the generating system of vectors is empty, i.e. does not contain any vectors, then by definition we assume that this vector space is zero, i.e. . in this case, by definition we assume that the basis of the zero vector space is an empty basis and its dimension is assumed to be zero.

Let further,  a non-zero vector space and a system of non-zero vectors  be its finite generating system.

If this system is linearly independent, then everything is proved, since the linearly independent and generating system of vectors of a vector space is its basis.

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

If the set A is linearly independent and covers R3, it means that it contains at least three independent vectors that are basis for R3.