Find the dimensions of the following vector spaces.
(a) The vector space ${\mathbb R}^{\,4}$
(b) The vector space of all diagonal $7 \times 7$ matrices
(c) The vector space of all lower triangular $6 \times 6$ matrices
(d) The vector space $P_2 [x]$ of polynomials with degree less than $2$
(e) The vector space of $7 \times 7$ matrices with trace 0
(f) The vector space ${\mathbb R}^{3 \times 5}$