linear independence: meaning, definition, pronunciation and examples

No. Orthogonal vectors are always linearly independent, but linearly independent vectors are not necessarily orthogonal. Orthogonality is a stronger geometric condition involving perpendicularity and zero dot product.

Yes, but only if it is not the zero vector. A set containing only the zero vector is linearly dependent.

The standard method is to form a matrix with the vectors as columns (or rows) and compute its determinant or row reduce it. If the determinant is non-zero (or row reduction yields a pivot in every column), the vectors are linearly independent. Alternatively, set up a linear combination equal to zero and solve; if the only solution is all zero coefficients, they are independent.

It is fundamental for defining the dimension of a vector space, constructing bases, solving systems of linear equations, and ensuring uniqueness in representations. In applied fields, it prevents redundancy in data and models.

A property of a set of vectors in a vector space where no vector in the set can be written as a linear combination of the others. Conceptually, it means the vectors provide unique information and there is no redundancy among them.

Linear independence is usually formal, academic, technical in register.

Linear independence: in British English it is pronounced /ˌlɪn.i.ər ˌɪn.dɪˈpen.dəns/, and in American English it is pronounced /ˌlɪn.i.ɚ ˌɪn.dɪˈpen.dəns/. Tap the audio buttons above to hear it.