A First Course in Linear Algebra · Problem VR-M10

Representations — Robert Beezer

Mathematics — Linear Algebra · coordinatization-basis-verification

Problem

Prove that the set

S

below is a basis for the vector space of

2 \times 2

matrices,

M_{22}

. Do this by choosing a natural basis for

M_{22}

and coordinatizing the elements of

S

with respect to this basis. Examine the resulting set of column vectors from

\mathbb{C}^4

and apply the Coordinatization Principle.

S = \left\{\begin{bmatrix} 33 & 99 \\ 78 & -9 \end{bmatrix}, \begin{bmatrix} -16 & -47 \\ -36 & 2 \end{bmatrix}, \begin{bmatrix} 10 & 27 \\ 17 & 3 \end{bmatrix}, \begin{bmatrix} -2 & -7 \\ -6 & 4 \end{bmatrix}\right\}