Let $\displaystyle {M}_{{1}}={\left[\begin{array}{ccc} {5}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{0}\end{array}\right]}$,$\displaystyle {M}_{{2}}={\left[\begin{array}{ccc} {0}&{0}&{0}\\{0}&-{2}&{0}\\{0}&{0}&-{4}\end{array}\right]}$

and $\displaystyle {M}_{{3}}={\left[\begin{array}{ccc} {0}&{0}&{0}\\{0}&-{3}&{0}\\{0}&{0}&{9}\end{array}\right]}$.

The set {$\displaystyle {M}_{{1}},{M}_{{2}},{M}_{{3}}$} spans the space of 3 X 3 diagonal matrices.

Now,$\displaystyle \ {D}={\left[\begin{array}{ccc} {20}&{0}&{0}\\{0}&-{34}&{0}\\{0}&{0}&{52}\end{array}\right]}$ can be written as a

linear combination of $\displaystyle {M}_{{1}},\ {M}_{{2}},\$and$\displaystyle \ {M}_{{3}}$.

That is,$\displaystyle \ {D}={a}_{{1}}{M}_{{1}}+{a}_{{2}}{M}_{{2}}+{a}_{{3}}{M}_{{3}}$.

Determine $\displaystyle {a}_{{1}}$,$\displaystyle {a}_{{2}}\$and $\displaystyle \ {a}_{{3}}$.

$\displaystyle {a}_{{1}}$=

$\displaystyle {a}_{{2}}$=

$\displaystyle {a}_{{2}}$=