Are Column Vectors in AB Unchanged if Linearly Independent?

In summary, if a 3x3 matrix AB of rank 2 is given and there is a request to find two linearly independent vectors in the column space of A, these vectors will also be linearly independent in the column space of AB. This is because if a vector is in the column space of AB, it can also be found in the column space of A. However, the reverse is not always true.
  • #1
only_huce
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If you have a 3x3 matrix AB of rank 2, and you are asked to find two linearly independent vectors in the column space of A, are those the same vectors in the column space of AB? I figure it is since if the vectors are linearly independent they wouldn't change. Am I right?
 
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  • #2
That's right. If x is in the column space of AB, then x = ABy for some vector y. But since x = A(By), x is in the column space of A. Thus, if you can find two linearly independent vectors in col AB, you've found two linearly independent vectors in col A.

Be careful, as the opposite isn't true: it's not necessarily true that if x is in the column space of A, then it's in the column space of AB.
 

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

Linear independence means that none of the vectors in a set can be written as a linear combination of the other vectors. In other words, each vector in the set is unique and cannot be expressed as a combination of the others.

2. How is linear independence related to the columns of a matrix?

The columns of a matrix represent the vectors in a set. If the columns are linearly independent, it means that each column vector is unique and cannot be expressed as a combination of the other columns.

3. Can a matrix have linearly dependent columns?

Yes, a matrix can have linearly dependent columns. This means that at least one of the column vectors can be expressed as a linear combination of the other columns.

4. How can I test if the columns of a matrix are linearly independent?

One way to test for linear independence is to use the determinant of the matrix. If the determinant is non-zero, then the columns are linearly independent. Another method is to use Gaussian elimination to check if the matrix can be reduced to an identity matrix.

5. What happens to the columns of a matrix when they are linearly independent?

If the columns of a matrix are linearly independent, they will remain unchanged when the matrix is multiplied by another matrix, such as in AB. This is because the linear independence ensures that each column vector is unique and cannot be expressed as a combination of the other columns.

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