A vector \(w\) is a linear combination of a set of vectors \(\{v_1, v_2, ..., v_k\}\) if it can be expressed as a sum of those vectors, each multiplied by a scalar constant.

Formula:

\(w = c_1v_1 + c_2v_2 + ... + c_kv_k\)

The constants \(c_1, c_2, ..., c_k\) are the scalars we need to find.

How it's Solved:

This problem can be rewritten as a system of linear equations. The vectors \(\{v_i\}\) form the columns of a matrix \(A\), the scalars \(\{c_i\}\) form a vector \(c\), and \(w\) is the target vector. We solve the matrix equation \(Ac = w\).

This calculator sets up the augmented matrix \([A|w]\) and uses Gauss-Jordan elimination to find the scalars in \(c\).