 # Check (2 1,0 0),(0 0,2 0),(3 -1,0 0),(0 3, 0 1) is a bais for M22 or not? Lewis Harvey 2020-12-21 Answered
Check $\left(21,00\right),\left(00,20\right),\left(3-1,00\right),\left(03,01\right)$ is a bais for M22 or not?
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Recall: Basis Let V be a vector space over a field F. A set of S of vectors in V is snid to be a basis of V if (i) S is linearly independent in V, and
(ii) S generates V ie. linear span of $S,L\left(S\right)=V$
Theorem: Let V be a vector space of dimension n over a field F. Then any linearly independent set of n vectors of V is a basis of V.
${M}_{22}$ is the vector space of all $2×2$ real matrices.
Let $S=\left\{\left(\begin{array}{c}21\\ 00\end{array}\right),\left(\begin{array}{c}00\\ 21\end{array}\right),\left(\begin{array}{c}3-1\\ 00\end{array}\right),\left(\begin{array}{c}00\\ 31\end{array}\right)\right\}$
We have to show that 5 is a basis of ${M}_{22}$
Let ${\alpha }_{1}=\left\{\left(\begin{array}{c}21\\ 00\end{array}\right),{\alpha }_{2}=\left(\begin{array}{c}00\\ 21\end{array}\right),{\alpha }_{3}=\left(\begin{array}{c}3-1\\ 00\end{array}\right),{\alpha }_{4}=\left(\begin{array}{c}00\\ 31\end{array}\right)\right\}$
Let us consider the relation
${c}_{1}{\alpha }_{1}+{c}_{2}{\alpha }_{2}+{c}_{3}{\alpha }_{3}+{c}_{4}{\alpha }_{4}=0$
where ${c}_{1},{c}_{2},{c}_{3},{c}_{4}\in RR.$
Then we have ${c}_{1}=\left(\begin{array}{c}21\\ 00\end{array}\right),+{c}_{2}\left(\begin{array}{c}00\\ 21\end{array}\right),+{c}_{3}\left(\begin{array}{c}3-1\\ 00\end{array}\right),+{c}_{4}\left(\begin{array}{c}00\\ 31\end{array}\right)=\left(\begin{array}{c}00\\ 00\end{array}\right)⇒\left(\begin{array}{c}2{c}_{1}+3{c}_{3}{c}_{1}-{c}_{3}\\ 2{c}_{2}+3{c}_{3}{c}_{2}+{c}_{4}\end{array}\right)=\left(\begin{array}{c}00\\ 00\end{array}\right)$
Fron the above equation we get,
$2{c}_{1}+3{c}_{3}=0$
${c}_{1}-{c}_{3}=0$
$2{c}_{2}+3{c}_{4}=0$
${c}_{2}+{c}_{4}=0$
Solving these equation we get, ${c}_{1}=0,{c}_{2}=0,{c}_{3}=0,{c}_{4}=0$
This proves that the set ${\alpha }_{1},{\alpha }_{2},{\alpha }_{3},{\alpha }_{4}$ is lineary independent.
Now, we know that ${M}_{22}$ is a vector space of dimension 4 and its standard
basis is $\left\{\left(\begin{array}{c}10\\ 00\end{array}\right),\left(\begin{array}{c}01\\ 00\end{array}\right),\left(\begin{array}{c}00\\ 10\end{array}\right),\left(\begin{array}{c}00\\ 01\end{array}\right)\right\}$
As the dimension of is a linearly independent
set containing 4 vectors of ${M}_{22},$
therefore the set ${\alpha }_{1},{\alpha }_{2},{\alpha }_{3},{\alpha }_{4}$
is a basis of ${M}_{2}2$.
Result:
$\left\{\left(\begin{array}{c}21\\ 00\end{array}\right),\left(\begin{array}{c}00\\ 21\end{array}\right),\left(\begin{array}{c}3-1\\ 00\end{array}\right),\left(\begin{array}{c}00\\ 31\end{array}\right)\right\}$
is a basis for ${M}_{22}$