Which rows are linearly dependent

Show 3 more comments. Active Oldest Votes. When you convert to RREF form, we look for "pivots" Notice that in this case, you only have one pivot.

A pivot is the first non-zero entity in a row. I asked about rows. In this instance it might be that the rows and columns have the same number of independent vectors, but is that always the case? If not, shouldn't we be framing the answer in terms of rows, not columns? So that would fit with what you're saying. But this book is also saying "linearly independent rows". Someone seems to think it makes sense to say "rows are independent". Add a comment.

Choose the largest such j. If not, then. Then we can rearrange:. If you make a set of vectors by adding one vector at a time, and if the span got bigger every time you added a vector, then your set is linearly independent.

Note that three vectors are linearly dependent if and only if they are coplanar. See this warning. In light of this important note and this criterion , it is natural to ask which columns of a matrix are redundant, i. The pivot columns are linearly independent, so we cannot delete any more columns without changing the span.

The following two vector equations have the same solution set, as they come from row-equivalent matrices:. Note that it is necessary to row reduce A to find which are its pivot columns.

However, the span of the columns of the row reduced matrix is generally not equal to the span of the columns of A : one must use the pivot columns of the original matrix. The indices of the zero elements in the null space show independence. But why is the third element here not zero? If we multiply the A matrix with the null space, we get a zero column vector. So what's wrong? Notice that the null space of a matrix is not equal to the null space of the transpose of that matrix unless it is symmetric.

Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Collectives on Stack Overflow. Learn more. How to find linearly independent rows from a matrix Ask Question. Asked 6 years, 8 months ago. Active 5 months ago.

Viewed 40k times. How to identify the linearly independent rows from a matrix? For instance, The 4th rows is independent. Improve this question. If I am not mistaken linear independent is a feature of a set of vectors.

I am not sure what identify the linearly independent rows means in this context. What exactly should be the output? Sorry for not expressing myself clearly. In this example, the rank is 3 so delete one of dependent rows say the 3rd row. Actually, the question what I want to track is here. Add a comment. Active Oldest Votes. Two methods you could use: Eigenvalue If one eigenvalue of the matrix is zero, its corresponding eigenvector is linearly dependent. Improve this answer.

Having went over vector spaces, we now combine the ideas of bases and dimension with the vector spaces, and the ideas of linear dependence and rank applied earlier. We will apply these important ideas to the general solution of m linear equations of n variables.

Consider the columns or rows of the matrix are elements of a vector space where addition and scalar multiplication were defined before. One can easily check that they form a vector space.