Exercise 1.2.1

Label the following statements as true or false.
(a) Every vector space contains a zero vector.
(b) A vector space may have more than one zero vector.
(c) In any vector space, $\mathit{ax}=\mathit{bx}$ implies that $a=b$.
(d) In any vector space, $\mathit{ax}=\mathit{ay}$ implies that $x=y$.
(e) A vector in $F^n$ may be regarded as a matrix in $M_{n\times 1}(F)$.
(f) An $m\times n$ matrix has $m$ columns and $n$ rows.
(g) In $P(F)$, only polynomials of the same degree may be added.
(h) If $f$ and $g$ are polynomials of degree $n$, then $f+g$ is a polynomial of degree $n$.
(i) If $f$ is a polynomial of degree $n$ and $c$ is a nonzero scalar, then $\mathit{cf}$ is a polynomial of degree $n$.
(j) A nonzero scalar of $F$ may be considered to be a polynomial in $P(F)$ having degree zero.
(k) Two functions in $\mathcal{ℱ}(S,F)$ are equal if and only if they have the same value at each element of $S$.