Mar 4, 2025 · I understand that the two pivot columns tell us which columns in our A matrix are linearly independent. I also understand that we can put this into the test equation for linear …

I have a very hard time remembering which is which between linear independence and linear dependence... that is, if I am asked to specify whether a set of vectors are linearly dependent …

Can the determinant (assuming it's non-zero) be used to determine that the vectors given are linearly independent, span the subspace and are a basis of that subspace?

Feb 23, 2017 · Any set of linearly independent vectors can be said to span a space. If you have linearly dependent vectors, then there is at least one redundant vector in the mix.

None of the columns are multiples of the others, but the columns do form a linearly dependent set. You know this without any real work, since $3$ vectors in $\mathbb {R}^2$ cannot form a …

Jan 24, 2016 · Old thread, but in fact putting the vectors in as columns and then computing reduced row echelon form gives you more insight about linear dependence than if you put …

Sep 5, 2015 · To me linearly independence is one of those math definitions that it is easier to get the feeling with characterisations (in this case something that isn't linearly dependent), rather …

However, every orthonormal set is linearly independent by the above theorem, as every orthonormal set is an orthogonal set consisting of nonzero vectors.

A set of vectors is linearly dependent when there are an infinite amount of solutions to the system of equations. This is non-trivial? Where does no solution come in? I understand that if there is …

Apr 3, 2018 · The n vectors are linearly dependent iff the zero vector is a nontrivial linear combination of the vectors (definition of linearly independent). The zero vector is a nontrivial …