I have a conceptual question with linear algebra. For deciding whether a set is

Question

I have a conceptual question with linear algebra. For deciding whether a set is a basis, I have to show that it spans the vector space and that is is linearly independent. Many of the solutions to my homework set up the coefficient matrix of the vectors and take a determinant. If the determinant is non zero, I only have the trivial solution and they say that this proves that it is a basis. How does only having the trivial solution prove span and linear independence? Can I always show that a set is a basis if I can make a coefficient matrix and show that the determinant is non zero? Thanks for the help

Explanation / Answer

As you know a set B is basis of V if

(1) B is linearly indpendent., And

(2) Each vector in V can be written as linear combination of vectors of B.

Now if you want to show that B is lineary independent then whenever combination of vectors in V equals 0,
All coefficient of vectors in linear combination must be 0.
Note that we will get a homogenous system of equation.
For this sometimes we get coefficient matrix as square matrix , then only we can calculate determinant ,If determinant is non zero we have only trivial solution otherwise non trivial solution exists.That is all coefficient is zero.
On the other hand if coefficient matrix is not square ,then we have to choose other method to determine the coefficients.

I hope it is clear to you!

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