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I'm having a really hard time with question b of the image below, (Find any non-trivial A and B such that they are orthogonal) and question c, the proof.

I know that non trivial means a nonzero solution, but how do I interpret the subscript? And does part c utilize Cauchy Schwarz? It looks almost like the pythagorean theorem but I'm not sure if I'm way off.

Any help or guidance you can provide would be super helpful, thanks! problem here

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    "Below"? Did you mean to attach something? – 2017-02-28
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    @JacobManaker yes, I thought it was showing up but maybe not -- I just updated it with a link at the bottom if it's not working! Thank you for pointing it out :) – 2017-02-28
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    For part (b): can you think of numbers such that $a_{11}b_{11} + a_{12}b_{12} + a_{22}b_{22} = 0$, but not all of the $a$'s are zero, and similarly for the $b$'s? hint: $1/2 + 1/2 + (-1) = 0$. For part (c): Have you heard of the "Cauchy-Schwarz inequality," perhaps during multivariable calculus? – 2017-02-28

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For b) we need only find $A$ and $B$ non-zero so that the inner product is $0$, i.e., $$ \langle A, B \rangle = \left\langle \begin{bmatrix} a_{11} & a_{12}\\ 0 & a_{22} \end{bmatrix}, \begin{bmatrix} b_{11} & b_{12}\\ 0 & b_{22} \end{bmatrix}\right\rangle = a_{11}b_{11} + a_{12}b_{12} + a_{22}b_{22} = 0$$ So one easy example would be taking $a_{11}=a_{12}=b_{22}=0$. This would give a sum of zeros, even if $A$ or $B$ were non-trivial. For instance,

$$ A= \begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}, \quad B= \begin{bmatrix} 3 & -17\\ 0 & 0 \end{bmatrix},\qquad \text{then $\langle A,B\rangle = 0$}$$

As for c), you are correct in recognizing the Cauchy-Schwartz inequality. As we have an inner product, it must satisfy $$\langle A, B\rangle^2 \leq \langle A, A\rangle \langle B,B\rangle$$ where $$\langle A, B\rangle^2 = (a_{11}b_{11}+a_{12}b_{12}+a_{22}b_{22})^2$$ $$\langle A, A\rangle \langle B,B\rangle = (a_{11}a_{11}+a_{12}a_{12}+a_{22}a_{22})(b_{11}b_{11}+b_{12}b_{12}+b_{22}b_{22}) = (a_{11}^2+a_{12}^2+a_{22}^2)(b_{11}^2+b_{12}^2+b_{22}^2)$$

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You can map each of this matrix to a vector of size 3. In each matrix there is a a zero entry. Ignore that zero and take the other three entries as components of a vector )that is the x,y,z co-ordinates)..

After this process it is clear that this question is not really about matrices, it is the same question as for $\mathbf{R}^3$.

If you have written computer programs the above process can be viewed this way. In a program we use, 2d-arrays, or 3d-arrays etc. A 2d-array can be scanned row by row and mapped to a 1-d array (that is what we have done).

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    Downvote because the relevance of your statements to the problem is unclear. – 2017-02-28
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    My answer was under the assumption the OP knows about inner products in $R^3$. Then my suggestion would make him see that matrices are irrelevant here. – 2017-02-28