can someone help me with the problem below? I know that I need to show it's linearly independent, but how do I rearrange those vectors into a matrix? Thanks so much!
How do I show that these vectors are a basis for $R^{2\times2}$?
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calculus
linear-algebra
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1Hint: show that every vector in the basis you know can be expressed as a linear combination of vectors from the set you must prove is itself a basis. For example: $$ \left[ \matrix{0 & 1 \\ 0 & 0} \right] = \left[ \matrix{1 & 1 \\ 0 & 0} \right] - \left[ \matrix{1 & 0 \\ 0 & 0} \right] $$ – 2017-01-17
1 Answers
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In your diagram, let us call the matrices in the standard basis(the first set of matrices given above) as $a,b,c,d$ in that order, and the matrices in the question (the second, or lower set of matrices) as $e,f,g,h$ in that order.. Then, I want you to see that: $$ a=e ; b = f-e ; c = h-g; d = g-f $$
The above shows that $a,b,c,d$ are all in the span of $e,f,g,h$. So, the span of $e,f,g,h$ contains the span of $a,b,c,d$, which is known to be $\mathbb R_{2 \times 2}$. Hence, the span of $e,f,g,h$ is also $\mathbb R_{2 \times 2}$.