MATH661

Quiz 2

True or false? $f (x) = a x + b$ , $f : ℝ \to ℝ$ is a linear mapping. Why?

Answer. True if and only if $b = 0$ . Verify linearity
$\forall x, y, u, v \in ℝ : f (u x + v y) = u f (x) + v f (y) \Rightarrow$ $a (u x + v y) + b = u (a x + b) + v (a y + b) \Rightarrow b = 0$
True or False? The 3x2 real-component matrix $𝑨$ can have maximal rank $r = 2$ .

Answer. True. Since $r = \dim C (𝑨)$ and $𝑨 \in ℝ^{3 \times 2}$ can have at most two linearly independent column vectors, e.g.,
$𝑨 = [\begin{array}{ll} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}] .$
True or false? $𝒙 = [\begin{array}{lll} 1 & 1 & 1 \end{array}]$ is an element of $C (𝑨)$ ,
$𝑨 = [\begin{array}{lll} 1 & 0 & 1 \\ 1 & - 1 & 0 \\ 0 & 0 & 0 \end{array}] .$
Explain.

Answer(s). False. Applying convention of column organization of vector components,
$𝒙 = [\begin{array}{l} 1 \\ 1 \\ 1 \end{array}] \notin C ([\begin{array}{lll} 1 & 0 & 1 \\ 1 & - 1 & 0 \\ 0 & 0 & 0 \end{array}]) .$
Imposing $𝒙 = 𝑨 𝒚$ leads to third-component equation
$1 = 0 \cdot y_{1} + 0 \cdot y_{2} + 0 \cdot y_{3},$
contradiction.

False. $𝒙$ is shown as a row vector.
True or false? It is known that $𝑨 (𝒙 + 𝒚) = 𝒃$ and $𝑨 𝒙 = 𝒃$ . Then $N (𝑨) = {𝟎}$ . Explain. Answer. Ambigous. Subtracting equations gives $𝑨 𝒚 = 𝟎$ , hence $𝒚 \in N (𝑨)$ . If $𝒚 = 𝟎$ then $N (𝑨) = {𝟎}$ . If $𝒚 \neq 𝟎$ then $N (𝑨) \neq {𝟎}$ .
True or False? $𝒙 \in N (𝑨)$ for
$𝒙 = [\begin{array}{l} 1 \\ 1 \\ 0 \end{array}], 𝑨 = [\begin{array}{lll} 0 & - 1 & 2 \\ 0 & 1 & - 2 \\ 1 & 0 & 2 \end{array}] .$
Answer. False. Compute
$𝑨 𝒙 = 1 \cdot [\begin{array}{l} 0 \\ 0 \\ 1 \end{array}] + 1 \cdot [\begin{array}{l} - 1 \\ 1 \\ 0 \end{array}] = [\begin{array}{l} - 1 \\ 1 \\ 1 \end{array}] \neq 𝟎 .$