Use the axioms of a vector space to prove that

$$(a+b)(u+v)=au+av+bu+bv$$

for all scalars $a$ and $b$ and all vectors $u$ and $v$ in a vector space.

Suppose that $W$ is a subspace of a vector space $V$. Prove that $W$ satisfies the axioms in the definition of vector space, and hence $W$ is itself a vector space.

Show that -0=0 in any vector space. Cite all axioms used.

Show that the $1-$norm satisfies the given properties of norms.

(a) $\|A\| \geq 0$ (b) $\|A\|=0$ if and only if $A=0$ (c) $\|c A\|=|c|\|A\|$ (d) $\|A+B\| \leq\|A\|+\|B\|$

These conditions are the axioms for a norm on a general vector space. Matrix norms are also required to satisfy the condition $\|A B\| \leq\|A\|\|B\|$. Show that the 1-norm also satisfies this condition.

Near Earth’s surface, the electric field has a magnitude of 150 N/C and points downward. Treating Earth’s surface as a conducting sphere, calculate the surface charge density necessary to produce this electric field.

Find a linearization of the differential equation

$$\frac{d^{2} x}{d t^{2}}+x e^{0.01 x}=0.$$

Find a basis of the space of solutions in R^n of the equation

$$x_1 + 2x_2 + 3x_3 + \ldots + nx_n = 0.$$

What is the empirical formula and empirical formula mass for each of the following compounds? C2H4

Apply a cusum control chart to the data in earlier exercise.

Two people exert the forces shown on the potted shrub. Determine the vector expression for the resultant R of the forces and determine the angle which the resultant makes with the positive y-axis

Both situations have the same expected value. Find the missing information. Situation 1: Paul hits a dart target worth 15 points 45% of the time. Situation 2: He hits a dart target worth 10 points x% of the time.

Graph each function. y=|3 x|

Again using Appendix F, which planet(s) might you expect not to have significant seasonal activity? Why?

A quadratic relation has roots 0 and -6 and a maximum at (-3, 4). Determine the equation of the relation.