Linear Independence of a 3x3 matrix

Linear independence means that no vector in the set can be written as a linear combination of the other vectors in the set. In this case, you need to show that u3 cannot be written as a linear combination of u1 and u2.
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bwilliams1188
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Homework Statement


A is a 3x3 matrix with distinct eigenvalues lambda(1), lambda(2), lambda(3) and corresponding eigenvectors u1,u2, u3.

Suppose you already know that {u1, u2} is linearly independent.

Prove that {u1, u2, u3} is linearly independent.


Homework Equations


??


The Attempt at a Solution


I am supposed to prove that {u1, u2, u3} is linearly independent, but since there are distinct eigenvalues/vectors, is that not enough to say that {u1, u2, u3} is linearly independent?
 
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  • #2
bwilliams1188 said:

Homework Statement


A is a 3x3 matrix with distinct eigenvalues lambda(1), lambda(2), lambda(3) and corresponding eigenvectors u1,u2, u3.

Suppose you already know that {u1, u2} is linearly independent.

Prove that {u1, u2, u3} is linearly independent.


Homework Equations


??


The Attempt at a Solution


I am supposed to prove that {u1, u2, u3} is linearly independent, but since there are distinct eigenvalues/vectors, is that not enough to say that {u1, u2, u3} is linearly independent?
No, that's not enough. That's exactly what you need to prove.
 

FAQ: Linear Independence of a 3x3 matrix

1. What is the definition of linear independence?

Linear independence refers to a set of vectors in a vector space that cannot be expressed as a linear combination of other vectors in the same space. In other words, none of the vectors in the set can be written as a combination of the others.

2. How can I determine if a set of vectors is linearly independent?

To determine linear independence, you can use the determinant method or the rank method. The determinant method involves finding the determinant of a matrix formed by the vectors, and if the determinant is non-zero, the vectors are linearly independent. The rank method involves putting the vectors into a matrix, reducing it to row echelon form, and checking the number of non-zero rows. If the number of non-zero rows is equal to the number of vectors, they are linearly independent.

3. What is the significance of linear independence in a 3x3 matrix?

In a 3x3 matrix, linear independence is important because it determines whether the matrix is invertible or not. If the matrix is linearly independent, it is invertible and has a unique solution. If the matrix is linearly dependent, it is not invertible and has either infinitely many solutions or no solutions at all.

4. Can a 3x3 matrix be linearly dependent?

Yes, a 3x3 matrix can be linearly dependent. This means that at least one of the vectors in the matrix can be written as a linear combination of the others. This results in a determinant of 0 and a rank less than 3, indicating that the matrix is not invertible.

5. Why is linear independence important in linear algebra?

Linear independence is important in linear algebra because it allows us to determine the invertibility of a matrix and the uniqueness of its solutions. It also plays a crucial role in solving systems of linear equations, diagonalizing matrices, and determining the basis of a vector space.

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