Are the columns linear independent?

In summary, the question is whether the columns of matrix A are linearly independent, given that it is in row reduced echelon form. The individual discussing the problem believes that the columns are dependent, based on the fact that the first two columns in their reduced forms are multiples of each other. However, they are questioning why the hint was given and if it is related to the definition of linear independence.
  • #1

Homework Statement



Were the columns of A linearly independent?
The matrix A is given in row reduced echeleon form
1 2 0 0
0 0 1 0
0 0 0 1


Homework Equations


Hint: Consider the solution set to A x = 0


The Attempt at a Solution


I think that the colums were linearly dependent since the first two columns in their row reduced forms are multiple of each other. Is that correct? If yes, and my explanation seems logical, then why is the hint given?
 
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  • #2
Perhaps they want you to see why "the first two columns in their row reduced forms are multiple of each other" implies that the columns of A are linearly dependent. In other words, the definition of linear independence says the vectors v1, v2, v3, and v4 are independent if the only solution of c1 v1+c2 v2+c3 v3+c4 v4=0 is c1=c2=c3=c4=0. How does what you said about the row-reduced matrix lead to the conclusion that the original columns of A are linearly dependent according to the definition above?
 

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

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.

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

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.

3. What are the implications of having linearly independent columns in a matrix?

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.

4. Can a matrix have both linearly independent and linearly dependent columns?

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.

5. Why is it important to check for linear independence in a matrix?

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.

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