Linear independence refers to a set of columns in a matrix being able to be combined in a linear combination to produce any vector in the vector space. In other words, no column in the set can be written as a linear combination of the other columns.
To determine if the columns in a matrix are linearly independent, you can use the determinant test. If the determinant of the matrix is non-zero, then the columns are linearly independent. Another method is to use Gaussian elimination to check if the columns can be reduced to a row of zeros.
If the columns are linearly independent, then the matrix is said to have full rank. This means that the columns span the entire vector space and can form a basis for the space. This also means that the matrix is invertible, and has a unique solution for every system of linear equations.
No, a matrix cannot have both linearly independent and linearly dependent columns. If even one column in a matrix is linearly dependent on the others, then the entire set of columns is linearly dependent. This means that the matrix does not have full rank and is not invertible.
Checking for linear independence in a matrix is important because it tells us if the matrix has full rank and is invertible. This is crucial in many applications, such as solving systems of linear equations, finding basis vectors, and performing dimensionality reduction. It also helps us understand the relationship between the columns in the matrix and the vector space they span.