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A matrix is defined as a rectangular array of numbers. Many of the arrays that you have worked have additional features, such as labels on the rows and columns or entries that are numbers with units. An array with additional features is technically not a matrix, instead we would call it a table. The difference between a matrix and a table is not vitally important in this unit, so we have worked with the underlying array of numbers and not worried about this technical detail. In more advanced courses in mathematics, like Linear Algebra, it will be important to be very precise about matrices. For now, describe and give examples of some of the ways a table can be different from a matrix. In your description and examples, be sure to include differences related to at least these three characteristics: entries, labels, and operations.

${\displaystyle (5x+3x^2-x^3)+(2x^2+4x-10)}$

Determine if the following series is convergent or divergent
a) $\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n\ln n}$
b) $\sum _{n=0}^{\mathrm{\infty }}n{e}^{-n^2}$
c) $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{\sqrt{n}}$
d)$\sum _{n=4}^{\mathrm{\infty }}\frac{1}{n^7}$

If John, Trey, and Miles want to know how’ | many two-letter secret codes there are that dont

Find the scalar and vector projections of b onto a
a=(-8,15) , b=(3,5)
Find the scalar and vector projections of b onto a
a=(6,7,-6) b=(3,-1,1)

Find the vectors, not with determinants, but by using properties of cross products.
$(i+j)\times (i-J)$
and
$k\times (i-2j)$

${\displaystyle \frac{{\left(\left(2x^3\right)y^2\right)}^{-3}}{{\left(\left(3x^2\right)y^4\right)}^{-2}}}$

Find a nonzero vector orthogonal to the plane through the points P, Q, and R, P(0,-2,0), Q(4,1,-2), R(5,3,1)

The line with y-intercept 4 and perpendicular to x+5y=8

Find the limit and discuss the continuity of the function. ${\displaystyle lim(x,y)\to (0,0)x+4y+1}$

How close to 0 do we need to take x so that
$(x^2+6x+9)<9.001?$

Insert one of the symbols << of >> to make the following statements true.
a) If ${\displaystyle x⟩}$, then x + 8 ___ 3.
b) If ${\displaystyle x⟩-3}$, then -5x + 18 ___ 3.

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number.
${\displaystyle f\left(x\right)=x^2+\sqrt{7-x},a=4}$

g is related to one of the parent functions described in Section 1.6. Describe the sequence of transformations from f to g. g(x) = √3x

Find derivatives of the functions defined as follows.
${\displaystyle y=-{10}^{3x^2-4}}$

One of the five given matrices represents an orthogonal projection onto a line and another represents a reflection about line. Identify both and briefly justify your choice.

Explain the term Comparable matrices?

Determine , if possible , the value of x which will male the following true:
$\left[\begin{array}{cc}4& 3\\ 1& x\end{array}\right]+2\left[\begin{array}{c}2\\ 1\end{array}\right]\left[\begin{array}{cc}3& x\end{array}\right]=\left[\begin{array}{cc}16& 11\\ 7& 6\end{array}\right]$

Let A and B be similar $n\times n$ matrices. Prove that if A is idempotent, then B is idempotent. (X is idempotent if $X^2=X$ )