Linearly Independent Columns of a Matrix

In summary, the conversation discusses the relationship between a matrix's determinant and the linear independence of its columns. It is stated that if the determinant is not equal to zero, then the columns are linearly independent because the matrix is invertible with maximal rank. This can be seen by converting the matrix into upper diagonal form and observing that the determinant is non-zero when the final matrix is diagonal with non-zero entries. This means that the columns can be made into 1's through matrix multiplication, proving their independence. This understanding helped the individual in their Linear Maths exam.
  • #1
sajama
5
0
Hi wondering if can anyone help me... I've gotten so bogged down in all the rules and stuff for singular/non-singular matrices I've completely confused myself!

Can anyone tell me is it true to say that if I have a matrix P, det(P) is NOT EQUAL to 0, then the vectors that would form the columns of P are linearly independent?

Cheers
 
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  • #2
Yes because then P is invertible, and thus have maximal rank.
 
  • #3
Brilliant - cheers! :)
 
  • #4
do you know why this is true? i.e. do you know what it really means that the determinant is not zero?

recall that any matrix can be rendered into upper diagonal form by repeatedly performing row operations, hence also by repeatedly multiplying by special invertible matrices.

then the determinant is non zero iff the final matrix is actually diagonal and has non zero entries on the diagonal. these can then be made 1's.

hence it has been inverted by matrix multiplication, and the columns are visibly independent, hence were also originally.
this is just one of many ways to see it.
 
  • #5
Thank you - it does make sense now - I had a Linear Maths exam this afternoon and even though at some stage I had understood the reasoning behind what made a matrix singular, my mind seemed to be blanking on me in the hours leading up to the exam!

Thanks again for the help :)
 

1. What does it mean for columns of a matrix to be linearly independent?

Linearly independent columns of a matrix means that no column in the matrix can be written as a linear combination of the other columns. In other words, the columns are not redundant and contribute unique information to the matrix.

2. How can I determine if the columns of a matrix are linearly independent?

To determine if the columns of a matrix are linearly independent, you can use the determinant of the matrix. If the determinant is non-zero, then the columns are linearly independent. Another way is to reduce the matrix to its echelon form and check if there are any rows of zeros, indicating linear dependence.

3. Why is it important for columns of a matrix to be linearly independent?

Linearly independent columns of a matrix are important because they allow us to solve for unique solutions to systems of equations. If the columns were linearly dependent, the system would have infinite solutions, making it impossible to solve.

4. Can a matrix have more linearly independent columns than rows?

Yes, a matrix can have more linearly independent columns than rows. This is known as an overdetermined system and often occurs in real-world applications where there are more variables than equations.

5. How does linear independence relate to the rank of a matrix?

The rank of a matrix is equal to the number of linearly independent columns in the matrix. This means that the rank can be used to determine if the columns of a matrix are linearly independent. If the rank is equal to the number of columns, then the columns are linearly independent.

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