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I'm having difficulty with how to begin with part (a), could someone please help me?
Thanks.

In the following question we regard vectors in $\Bbb R^n$ as columns and define the dot product in the usual way which means that $x\cdot y = x^ty$

(a) If A is an n x n matrix show that $A_{ij}= e_i \cdot Ae_j$ where $e_i, i= 1,...,n$ are the standard basis vectors in $\Bbb R^n$.

(b) Show that an n x n matrix P is orthogonal if and only if $(Px) \cdot (Py) = x \cdot y $ for all x, y in $\Bbb R^n$

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    $\begingroup$ Hint: try doing the multiplication for particular basis vectors. $\endgroup$ – Sean Roberson Oct 2 '17 at 9:27
  • $\begingroup$ The product of $A$ and $e_j$ is the $j^{\text{th}}$ column vector of $A$. Now the product of any vector with $e_i$ gives its $i^{\text{th}}$ coordinate, in our case $A_{ij}$. $\endgroup$ – Michael Hoppe Oct 2 '17 at 10:00
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Only using the definition of matrix product, you should find: $$(Ae_j)_{k1}=\sum_{l=1}^nA_{kl}(e_j)_{l1}=A_{kj}$$

and

$$(e_i^tAe_j)_{11}=\sum_{l=1}^n(e_i^t)_{1l}(Ae_j)_{l1}=\sum_{l=1}^n(e_i^t)_{1l}A_{lj}=A_{ij}.$$

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