RRt or RtR is always symmetric because for any element i j (row i of r and column j of rt) is the resulting matrix the corresponding ji (row j of r and column i of rt) element

row R of r equals column R of rt and column C of r equals row R of rt

(row i of r and column j of rt) = row i of r * row j of r

(row j of r and column i of rt) = row j of r and row i of r hence the resulting matrix is symmetric

simple proof )(RRt )T= RttRt= RRt hence symmetric

vector spaces are closed under addition and multiplication ie linear combinations of the vectors