Question

Which of the following statements are true ?
Give reasons for your answers.
(a) If A belongs to Mn(R), then rank (A)=det(A).
(b) There is one and only one unitary matrix in
Mn(R).
(c) If U and V are subspaces of a vector space
W over R, then U U V is also a subspace of W.
(d) Given a linear transformation T from
R4 to R6, rank (T) + nullity (T) = 6.
(e) The relation R, defined on the set of lines in
R² by `L1 R L2 iff L1 and L2 intersect', is an
equivalence relation.

Answer 1

a) False. The matrix $\begin{pmatrix} 1&0\\0&1 \end{pmatrix}$ is of rank 2 and determinant 1.

b) False. For example, $\begin{pmatrix} 1&0\\0&1 \end{pmatrix}$ and $\begin{pmatrix} 0&1\\1&0 \end{pmatrix}$ are unitary matrices.

c) False. For example, take $U$ to be the $x$-axis and $V$ $y$-axis, both subspaces of $\mathbb{R}^2$. Their union includes both (1,0) and (0,1), whose sum, (1,1), is not in the union. Hence, the union is not a vector space.

d) False. According to Rank-Nullity theorem, $\text{rank}(T)+\text{nullity}(T)=\dim(\mathbb{R^4})=4\neq6.$

e) False. The relation $R$ is not transitive. For example, take $L_1,\;L_2\text{ and } L_3$ are the lines of equations $y=x,\;y=-x\text{ and }y=x-1$ respectively. We have

$$L_1\,R \,L_2\text{ and }L_2\, R\, L_3,\text{ but }L_1\not R\, L_3.$$