Let A and C be the following ordered bases of P3(t): A = (1, 1 + t, 1 + t + t^2, 1 + t + t^2 + t^3) C = (1, -t, t^2 -t^3) Find tha change of coordinate matrix I_{CA}

To determine:
Find the sets of points in space whose coordinates satisfy the given combinations of equation and inequalities:
a) ${\displaystyle y\ge x^2,z\ge 0,}$
b) ${\displaystyle x\le y^2,0\le z\le 2.}$

Given the elow bases for $R^2$ and the point at the specified coordinate in the standard basis as below, (40 points)

$(B1=\{(1,0),(0,1)\}$&

$B2=(1,2),(2,-1)\}$(1, 7) = $3^{\ast }(1,2)-(2,1)$
$B2=(1,1),(-1,1)(3,7=5^{\ast }(1,1)+2^{\ast }(-1,1)$
$B2=(1,2),(2,1)(0,3)=2^{\ast }(1,2)-2^{\ast }(2,1)$
$(8,10)=4^{\ast }(1,2)+2^{\ast }(2,1)$
B2 = (1, 2), (-2, 1) (0, 5) =
(1, 7) =

a. Use graph technique to find the coordinate in the second basis. (10 points) b. Show that each basis is orthogonal. (5 points) c. Determine if each basis is normal. (5 points) d. Find the transition matrix from the standard basis to the alternate basis. (15 points)

 

To determine:
a) Whether the statement, " The point with Cartesian coordinates ${\displaystyle \left[\begin{array}{cc}-2& 2\end{array}\right]}$ has polar coordinates ${\displaystyle \left[bf(2\sqrt{2},\frac{3\pi }{4})(2\sqrt{2},\frac{11\pi }{4})(2\sqrt{2},-\frac{5\pi }{4})\text{and}(-2\sqrt{2},-\frac{\pi }{4})\right]}$ " is true or false.
b) Whether the statement, " the graphs of ${\displaystyle [r\cos \theta =4\text{and}r\sin \theta =-2]}$ intersect exactly once " is true or false.
c) Whether the statement, " the graphs of ${\displaystyle [r=4\text{and}\theta =\frac{\pi }{4}]}$ intersect exactly once ", is true or false.
d) Whether the statement, " the point ${\displaystyle \left[\begin{array}{cc}3& \frac{\pi }{2}\end{array}\right]liesonthegraphof[r=3\cos 2\theta ]}$ " is true or false.
e) Whether the statement, " the graphs of ${\displaystyle [r=2\sec \theta \text{and}r=3\csc \theta ]}$ are lines " is true or false.

The volume of the largest rectangular box which lies in the first octant with three faces in the coordinate planes and its one of the vertex in the plane ${\displaystyle x+2y+3z=6}$ by using Lagrange multipliers.

Since we will be using various bases and the coordinate systems they define, let's review how we translate between coordinate systems. a. Suppose that we have a basis${\displaystyle B=\{v_1,v_2,\dots ,v_m\}f\text{or}{R}^m}$. Explain what we mean by the representation {x}g of a vector x in the coordinate system defined by B. b. If we are given the representation ${\displaystyle {\left\{x\right\}}_B,}$ how can we recover the vector x? c. If we are given the vector x, how can we find ${\displaystyle {\left\{x\right\}}_B}$? d. Suppose that BE is a basis for R^2. If $x_B=\left[\begin{array}{c}1\\ -2\end{array}\right]$ find the vector x. e. If $x=\left[\begin{array}{c}2\\ -4\end{array}\right]find{x}_B$

What are polar coordinates? What equations relate polar coordi-nates to Cartesian coordinates? Why might you want to change from one coordinate system to the other?

The Cartesian coordinates of a point are given.
a) $(2,-2)$
b) $(-1,\sqrt{3})$
Find the polar coordinates $(r,\theta )$ of the point, where r is greater than 0 and 0 is less than or equal to $\theta$, which is less than $2\pi$
Find the polar coordinates $(r,\theta )$ of the point, where r is less than 0 and 0 is less than or equal to $\theta$, which is less than $2\pi$