A First Course in Linear Algebra · Problem RQ-LISS-3

Vector Spaces — Robert Beezer

Mathematics — Linear Algebra · vector-representation-basis

Problem

The set

S = \left\{\begin{bmatrix}6\\2\\1\end{bmatrix},\begin{bmatrix}4\\-3\\1\end{bmatrix},\begin{bmatrix}5\\8\\2\end{bmatrix}\right\}

is linearly independent and spans

\mathbb{C}^3

. Write the vector

\mathbf{x} = \begin{bmatrix}-6\\2\\2\end{bmatrix}

as a linear combination of the elements of

S

. How many ways are there to answer this question, and which theorem allows you to say so?