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 or …

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?

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 no …

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 independence.

12 you can take the vectors to form a matrix and check its determinant. If the determinant is non zero, then the vectors are linearly independent. Otherwise, they are linearly dependent.

The vectors are linearly independent if the only linear combination of them that's zero is the one with all $\alpha_i$ equal to 0. It doesn't make sense to ask if a linear combination of a set of vectors (which …

Jan 9, 2021 · I assume this is because the determinant encodes a sort of "test" for linear independence, so that instead of determining if, for example, three vectors are linearly independent with the definition:

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 linear combination of …

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 than …

Oct 4, 2017 · Well i'm reading in a book that the rank of a matrix is equal to the maximum number of linearly independent rows or, equivalently, the dimension of the row space. So does that mean …