Related questions with answers

Define $T:R^3\to R^3$ by

\begin{equation*}T\begin{pmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\end{pmatrix}=\begin{bmatrix}0\\0\\x_3\end{bmatrix}\end{equation*}

which is the projection of

\begin{equation*}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\end{equation*}

on the z-axis. (a) What is the null space of T? (b) Is T one-to-one? (c) What is the range of T? (d) Is T onto?

(a) The mobility of a $\mathrm{Rb}^{+}$ ion in aqueous solution is $792 \times 10^{-4} \mathrm{m}^{2} \mathrm{s}^{-1} \mathrm{V}^{-1}$ at $25^{\circ} \mathrm{C}.$ The potential difference between two electrodes, separated by 7.00 mm and placed in the solution, is 25.0 V. What is the drift speed of the $\mathrm{Rb}^{+}$ ion? (b) The mobility of a $\mathrm{Li}^{+}$ ion in aqueous solution is $4.01 \times 10^{-4} \mathrm{m}^{2} \mathrm{s}^{-1} \mathrm{V}^{-1}$ at $25^{\circ} \mathrm{C}.$ The potential difference between two electrodes separated by 5.00 mm and placed in the solution, is 24.0 V. What is the drift speed of the ion?

In gel electrophoresis, the mobility $\mu$ of a molecule in a particular gel matrix is defined as $\mu = v _ { \mathrm { f } } / E$ , where $v _ { \mathrm { t } }$ is the terminal speed of the molecule and E is the applied electric field strength. In one case, a molecule has mobility $3.0 \times 10 ^ { - 8 } \mathrm { C } \cdot \mathrm { m } / ( \mathrm { N } \cdot \mathrm { s } )$ and charge -12e. (a) Estimate the electric field that should be applied to give this molecule a terminal speed of $2.0 \times 10 ^ { - 5 }$ m/s. (b) How long does it take the molecule to move 2.0 cm through the gel? (c) Suppose the same molecule had a charge of -8e instead of -12 e. Considering the forces exerted on the molecule, would its terminal speed be smaller or larger (for the same applied field)? Would its mobility be smaller or larger?