# Proving Linear Independence of Non-Zero Rows in Row-Echelon Form

• Benny
In summary: The matrix is a representation of the set of nonzero row vectors. The position of the vectors is not important.
Benny
Hi, can someone help me with the following question?

Q. Show that if $$\left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to } \right\}$$ is linearly independent and $$\mathop {v_{k + 1} }\limits^ \to \notin span\left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to } \right\}$$ then $$\left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to ,\mathop {v_{k + 1} }\limits^ \to } \right\}$$ is linearly independent. Use this to prove that the non-zero rows of a matrix in row-echelon form are linearly independent.

Here is my attempt.

Write $$\alpha _1 \mathop {v_1 }\limits^ \to + ... + \alpha _k \mathop {v_k }\limits^ \to + \beta \mathop {v_{k + 1} }\limits^ \to = \mathop 0\limits^ \to ...\left( 1 \right)$$

$$\beta \mathop {v_{k + 1} }\limits^ \to = - \left( {\alpha _1 \mathop {v_1 }\limits^ \to + ... + \alpha _k \mathop {v_k }\limits^ \to } \right)$$

If $$\beta \ne 0$$ then $$\mathop {v_{k + 1} }\limits^ \to = - \left( {\frac{{\alpha _1 }}{\beta }\mathop {v_1 }\limits^ \to + ...\frac{{\alpha _k }}{\beta }\mathop {v_k }\limits^ \to } \right)$$ but this is impossible since $$\mathop {v_{k + 1} }\limits^ \to \notin span\left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to } \right\}$$

So beta is equal to zero and equation one reduces to $$\alpha _1 \mathop {v_1 }\limits^ \to + ... + \alpha _k \mathop {v_k }\limits^ \to = \mathop 0\limits^ \to$$ where all of the a_i are equal to zero by hypothesis. Is that enough to show the given result?

I can't think of a way to tackle the second part with the matrix. Seeing as that's the case I'll just write out whatever I can think of.

I think the key idea is that in row echelon form, each time I 'move up' one row, the vector(represented by a row in the matrix) has at least one additional non-zero component. So let A be the n by k (n columns and k rows) matrix in row echelon form whose rows are the vectors v_i where i = 1,...,k and each of the vectors has at least one non-zero component.

Starting at the bottom of the matrix and moving up to the first non-zero row I a vector which has c non-zero components call it v_1 and {(v_1)} is linearly independent since it consists of a non-zero single vector. Moving up to the next row I get another vector call it v_2 which has at least c + 1 non-zero components. Since v_2 has more non-zero components than v_1 then {v_1, v_2} is linearly independent. From here I'd probably just continue with the same argument. The problem is that what I've said is a pretty clumsy explanation. I wasn't really sure how to do this question either. So can someone please help me with this?

Edit: Ok my attempt for the second part is completely incorrect because I could have something like v_1 = (0,0,1,0,0) and v_2 = (1,0,0,0,0). Help would be appreciated.

Last edited:
Do the second part by induction.
n=1 case
the vector is not zero so is linearly independent
n+1 case
n vectors are independent
if (n+1)st vector is in span(first n vectors) the matrix is not in row-echelon form
therfore (n+1)st vector is not in span
thus n+1 vectors are linearly independent

Thanks for te help lurflurf. However, I am not sure how my answer should be worded. The question refers to a matrix so would I need to make some reference to the matrix? If so how would I do it in a clear and concise manner? For example, do I need to mention the position of the vectors(represented as rows) in the matrix?

Benny said:
Thanks for te help lurflurf. However, I am not sure how my answer should be worded. The question refers to a matrix so would I need to make some reference to the matrix? If so how would I do it in a clear and concise manner? For example, do I need to mention the position of the vectors(represented as rows) in the matrix?
The matrix is a representation. You can consider the set of the nonzero row vectors. The position of the vectors is not important.
Just say something like
let v1,...,vn be the nonzero row vectors

## What is the definition of linear independence?

Linear independence refers to a set of vectors in a vector space where no vector can be expressed as a linear combination of the other vectors in the set.

## Why is it important to prove linear independence of non-zero rows in row-echelon form?

Proving linear independence of non-zero rows in row-echelon form is important because it shows that the system of equations represented by the rows has a unique solution. This is essential in solving many real-world problems in fields such as physics, engineering, and economics.

## How do you prove linear independence of non-zero rows in row-echelon form?

To prove linear independence of non-zero rows in row-echelon form, you can use the reduced row-echelon form of the matrix and check for any rows that are all zero or contain only one non-zero element. If there are no such rows, then the non-zero rows in the original row-echelon form are linearly independent.

## What are the implications of a set of vectors being linearly dependent?

If a set of vectors is linearly dependent, it means that at least one vector in the set can be expressed as a linear combination of the other vectors. This can lead to inconsistencies and multiple solutions in a system of equations, making it difficult to find a unique solution.

## Is there a quick way to check for linear independence of non-zero rows in row-echelon form?

Yes, there is a quick way to check for linear independence of non-zero rows in row-echelon form. You can use the determinant of the matrix formed by the non-zero rows. If the determinant is non-zero, then the rows are linearly independent. However, if the determinant is zero, then further analysis is needed to determine the linear independence of the rows.

• Linear and Abstract Algebra
Replies
3
Views
976
• Linear and Abstract Algebra
Replies
8
Views
832
• Topology and Analysis
Replies
2
Views
2K
• Topology and Analysis
Replies
2
Views
2K
• Linear and Abstract Algebra
Replies
10
Views
1K
• Linear and Abstract Algebra
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
881
• Introductory Physics Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
963