Given that (p = begin{pmatrix} 5 \ 3 end{pmatrix}, q = begin{pmatrix} -1 \ 2 end{pmatrix}) and (r = begin{pmatrix} 17 \ 5 end{pmatrix}) and (r = alpha r + beta q), where (alpha) and (beta) are scalars, express q in terms of r and p.
Explanation
(p = begin{pmatrix} 5 \ 3 end{pmatrix}; q = begin{pmatrix} -1 \ 2 end{pmatrix}; r = begin{pmatrix} 17 \ 5 end{pmatrix})
(r = alpha p + beta q)
(begin{pmatrix} 17 \ 5 end{pmatrix} = alpha (begin{pmatrix} 5 \ 3 end{pmatrix}) + beta (begin{pmatrix} -1 \ 2 end{pmatrix})
(17 = 5alpha – beta … (1); 5 = 3alpha + 2beta …. (2))
((1) times 2 : 34 = 10alpha – 2beta … (3))
((3) + (2) : 39 = 13alpha implies alpha = 3)
(5 = 3(3) + 2beta implies 2 beta = 5 – 9 = -4)
(beta = -2)
(r = 3p – 2q implies 2q = 3p – r)
(q = frac{1}{2} (3p – r))