Keyword Analysis & Research: when are columns linearly independent

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How to determine if a vector set is linearly independent?

set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero The set is of course dependent if the determinant is zero. Example The vectors <1,2> and <-5,3> are linearly independent since the matrix has a non-zero determinant. Example

Does linearly independent imply all elements are orthogonal?

linearly independent (Adjective) (Of a set of vectors or ring elements) whose nontrivial linear combinations are nonzero. Does linearly independent imply all elements are orthogonal? Vectors which are orthogonal to each other are linearly independent. But this does not imply that all linearly independent vectors are also orthogonal.

When are matrices linearly independent?

What makes a matrix linearly independent? If the determinant is not equal to zero, it’s linearly independent. Otherwise it’s linearly dependent. Since the determinant is zero, the matrix is linearly dependent. Are dependent matrices invertible? If A is a square matrix with linearly dependent columns, then A is not invertible.

Can a single vector be linearly independent?

Note that because a single vector trivially forms by itself a set of linearly independent vectors. Moreover, because otherwise would be linearly independent, a contradiction. Now, can be written as a linear combination of : where are scalars and they are not all zero (otherwise would be zero and hence not an eigenvector).

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