A Iinear transformation T, defined as \[T\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ {{x_3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{x_1} + {x_2}} \\ {{x_2} - {x_3}} \end{array}} \right],\] transforms a vector \[\overrightarrow x \] for a three-dimensional real space to a two-dimensional real space. The transformation matrix T is
A. \[\left[ {\begin{array}{*{20}{c}} 1&1&0 \\ 0&1&{ - 1} \end{array}} \right]\]
B. \[\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \end{array}} \right]\]
C. \[\left[ {\begin{array}{*{20}{c}} 1&1&1 \\ { - 1}&1&1 \end{array}} \right]\]
D. \[\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&0&1 \end{array}} \right]\]
Answer: Option A
A. $$\frac{{1 + i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 + i}}{{\sqrt 2 }}a$$
B. $$ia{\text{ and }} - ia$$
C. $$ia,\, - ia,\,\frac{{1 - i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 - i}}{{\sqrt 2 }}a$$
D. $$\frac{{1 + i}}{{\sqrt 2 }}a,\, - \frac{{1 + i}}{{\sqrt 2 }}a,\,\frac{{1 - i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 - i}}{{\sqrt 2 }}a$$
Which of the following functions of the complex variable z is not analytic everywhere?
A. ez
B. $$\sin \frac{{\text{z}}}{{\text{z}}}$$
C. e3
D. |z|3
A. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + 3{{{\bf{\hat j}}}^{\bf{'}}} + \left( {1 + \sqrt 3 } \right){{{\bf{\hat k}}}^{\bf{'}}}\]
B. \[\left( {1 + \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + 3{{{\bf{\hat j}}}^{\bf{'}}} + \left( {1 - \sqrt 3 } \right){{{\bf{\hat k}}}^{\bf{'}}}\]
C. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + \left( {3 + \sqrt 3 } \right){{{\bf{\hat j}}}^{\bf{'}}} + 2{{{\bf{\hat k}}}^{\bf{'}}}\]
D. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + \left( {3 - \sqrt 3 } \right){{{\bf{\hat j}}}^{\bf{'}}} + 2{{{\bf{\hat k}}}^{\bf{'}}}\]
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