Explained: "Check (21,00),(00,20),(3−1,00),(03,01) is a bais for M22 or..."

College
Algebra
Linear algebra
Matrix transformations

Lewis Harvey

Answered question

2020-12-21

Check

(21, 00), (00, 20), (3 - 1, 00), (03, 01)

is a bais for M22 or not?

Answer & Explanation

liingliing8

Skilled2020-12-22Added 95 answers

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 $S, L (S) = V$
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 $2 \times 2$ real matrices.
Let $S = {(\begin{matrix} 21 \\ 00 \end{matrix}), (\begin{matrix} 00 \\ 21 \end{matrix}), (\begin{matrix} 3 - 1 \\ 00 \end{matrix}), (\begin{matrix} 00 \\ 31 \end{matrix})}$
We have to show that 5 is a basis of $M_{22}$
Let $α_{1} = {(\begin{matrix} 21 \\ 00 \end{matrix}), α_{2} = (\begin{matrix} 00 \\ 21 \end{matrix}), α_{3} = (\begin{matrix} 3 - 1 \\ 00 \end{matrix}), α_{4} = (\begin{matrix} 00 \\ 31 \end{matrix})}$
Let us consider the relation
$c_{1} α_{1} + c_{2} α_{2} + c_{3} α_{3} + c_{4} α_{4} = 0$
where $c_{1}, c_{2}, c_{3}, c_{4} \in R R .$
Then we have $c_{1} = (\begin{matrix} 21 \\ 00 \end{matrix}), + c_{2} (\begin{matrix} 00 \\ 21 \end{matrix}), + c_{3} (\begin{matrix} 3 - 1 \\ 00 \end{matrix}), + c_{4} (\begin{matrix} 00 \\ 31 \end{matrix}) = (\begin{matrix} 00 \\ 00 \end{matrix}) \Rightarrow (\begin{matrix} 2 c_{1} + 3 c_{3} c_{1} - c_{3} \\ 2 c_{2} + 3 c_{3} c_{2} + c_{4} \end{matrix}) = (\begin{matrix} 00 \\ 00 \end{matrix})$
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 $α_{1}, α_{2}, α_{3}, α_{4}$ is lineary independent.
Now, we know that $M_{22}$ is a vector space of dimension 4 and its standard
basis is ${(\begin{matrix} 10 \\ 00 \end{matrix}), (\begin{matrix} 01 \\ 00 \end{matrix}), (\begin{matrix} 00 \\ 10 \end{matrix}), (\begin{matrix} 00 \\ 01 \end{matrix})}$
As the dimension of $M_{22} i s 4 a n d α_{1}, α_{2}, α_{2}, α_{4}$ is a linearly independent
set containing 4 vectors of $M_{22},$
therefore the set $α_{1}, α_{2}, α_{3}, α_{4}$
is a basis of $M_{2} 2$ .
Result:
${(\begin{matrix} 21 \\ 00 \end{matrix}), (\begin{matrix} 00 \\ 21 \end{matrix}), (\begin{matrix} 3 - 1 \\ 00 \end{matrix}), (\begin{matrix} 00 \\ 31 \end{matrix})}$
is a basis for $M_{22}$

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