Invertability as a substitute for looking for linear independence

[ichigo444]

Can i use the invertability of a matrix as an alternative way of determining the linear independence of a set? Thank you.

 

[Mark44]

Yes. If the columns of the matrix are considered to be vectors, these vectors are linearly independent iff the matrix is invertible.

 

[Wizlem]

If you suppose the set creates a matrix A, then you can say the set is linearly independent if Ax = 0 has only the trivial solution. If A is invertible then x = A^-1 0 = 0 is the only solution and the set is linearly independent.

 

[Wizlem]

Yes. If the columns of the matrix are considered to be vectors, these vectors are linearly independent iff the matrix is invertible.

This is true for a square matrix. A non-square matrix is obviously not invertible but can be made of linearly independent vectors.