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Find a basis for the eigenspace corresponding to each listed eigenvalue of A below.
$$A=\left[\begin{array}{cc}1& 0\\ -1& 2\end{array}\right],\lambda =2,1$$

Find a least-squares solution of Ax=b by (a) constructing the normal equations for ${\displaystyle \hat{x}}$ and (b) solving for ${\displaystyle \hat{x}}$
$$A=\left[\begin{array}{cc}2& 1\\ -2& 0\\ 2& 3\end{array}\right],b=\left[\begin{array}{c}-5\\ 8\\ 1\end{array}\right]$$

Summarize how graphs can be used to find solutions to polynomial inequalities.

Please, help toconvert the Polar Equation to Cartesian Coordinates:
${\displaystyle r^2=\sec 4\theta }$

The relationship between f and g is given. Explain the relationship between f′ and g′
${\displaystyle g\left(x\right)=2f\left(x\right)}$

sin(theta) is opposite/hypotenuse regarding right triangles. Then what would ${\displaystyle \sin \left(2\theta \right)}$ be?

If ${\displaystyle f\left(x\right)=\cos \left(2x\right),thenf{}^{‴}\left(\frac{\pi }{4}\right)=}$?

Use the given of the coefficient matrix to solve the following system.
${\displaystyle 7x_1+3x_2=6}$
${\displaystyle -6x_1-3x_2=4}$
$$A^{-1}=\left[\begin{array}{cc}1& 1\\ -2& -\frac{7}{3}\end{array}\right]$$

Let ${\displaystyle R_1}$ and ${\displaystyle R_2}$ be a relations on a set A represented by the matrices below:
$$M_{R1}=\left[\begin{array}{ccc}0& 1& 0\\ 1& 1& 1\\ 1& 0& 0\end{array}\right]$$
$$M_{R2}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 1& 1\\ 1& 1& 1\end{array}\right]$$
Find the matrix that represents ${\displaystyle R_1\cap R_2}$

Find A such that the given set is Col A.
$$\{\left[\begin{array}{c}2s+3t\\ r+s-2t\\ 4r+s\\ 3r-s-t\end{array}\right]:r,s,treal\}$$

Find a function f such that ${\displaystyle f^{\prime }\left(x\right)=3x^3}$ and the line 81x+y=0 is tangent to the graph of f.

Tell the opposite to a cross product.
For example, ${\displaystyle a\times b=c}$. How to find c, if you know only a and b?

Find the exact length of the curve.
${\displaystyle x=e^t+e^{-t},y=5-2t,0\le t\le 3}$

Solve the initial value problem for
${\displaystyle \mathbb{r}}$
as a vector function of t. Differential equation:
${\displaystyle \frac{d^2\mathbf{r}}{{dt}^2}=-32\mathbf{k}}$
Initial conditions:
r(0)=100k and
${\displaystyle \frac{dr}{dt}{\mid }_{t=0}=8i+8j}$

Find a unit vector that has the same direction as the given vector. -3i + 7j

For the matrices (a) find k such that Nul A is a subspace of ${\displaystyle {\mathbb{R}}^k}$, and (b) find k such that Col A is a subspace of ${\displaystyle {\mathbb{R}}^k}$
$$A=\left[\begin{array}{ccccc}4& 5& -2& 6& 0\\ 1& 1& 0& 1& 0\end{array}\right]$$

Taylor’s formula with ${\displaystyle n=1}$ and ${\displaystyle a=0}$ gives the linearization of a function at ${\displaystyle x=0}$ With ${\displaystyle n=2}$ and ${\displaystyle n=3}$ we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions:
a) For what values of x can the function be replaced by each approximation with an error less than ${\displaystyle {10}^{-2}}$
b. What is the maximum error we could expect if we replace the function by each approximation over the specified interval? Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and interval.
Step 1: Plot the function over the specified interval.
Step 2: Find the Taylor polynomials ${\displaystyle P_1\left(x\right),P_2\left(x\right)\text{and}{P}_3\left(x\right)}$ at ${\displaystyle x=0}$
Step 3: Calculate the ${\displaystyle (n+1)}$ st derivative ${\displaystyle f^{(n+1)}\left(c\right)}$ associated with the remainder term for each Taylor polynomial. Plot the derivative as a function of c over the specified interval and estimate its maximum absolute value, M.
Step 4: Calculate the remainder ${\displaystyle R_n\left(x\right)}$ for each polynomial. Using the estimate M from Step 3 in place of ${\displaystyle f_{(n+1)}\left(c\right)}$ plot ${\displaystyle R_n\left(x\right)}$ over the specified interval. Then estimate the values of x that answer question (a).
Step 5: Compare your estimated error with the actual error ${\displaystyle E_n\left(x\right)=|f\left(x\right)-P_n\left(x\right)|}$ by plotting ${\displaystyle E_n\left(x\right)}$ over the specified interval. This will help answer question (b).
Step 6: Graph the function and its three Taylor approximations together. Discuss the graphs in relation to the information discovered in Steps 4 and 5.
${\displaystyle f\left(x\right)={(1+x)}^{\frac{3}{2}},-\frac{1}{2}\le x\le 2}$

Identify the surface with the given vector equation.
${\displaystyle r(s,t)=<s\sin 2t,s^2,s\cos 2t>}$

Suppose that f(x) = x/8 for 3 < x < 5. Determine the following probabilities: a. P(X < 4) b. P(X > 3.5) c. P(4 < X < 5) d. P(X < 4.5) e. P(X < 3.5 or X > 4.5)