Vector times symmetric matrix For most of you here, this is probably quite basic.
As for a symmetric matrix $A$ the first row equals the first column, multiplying the matrix with a column vector $b$ equals multiplying the transposed vector $b'$ with the symmetric matrix, i.e. if $A=A'$ then
$$Ab=b'A$$
Could you please confirm? Is there a better derivation than the verbal one above?
Also, while I have found many sources online about matrix algebra, I have not found this property, yet.  is there a reliable source on the web where one could find this? I was hoping that such a site would also contain additional information which might help to answer some other questions I have.

Except in dimension $1$, your claim is not correct. However, because transposition swaps factors we have
$$ (Ab)^T=b^TA^T=b^TA$$
that is multiplying with a column vector from the right equals the transpose of multiplying the transposed vector from the left.
