Simplifying Linear Algebra: Techniques, Solutions, and Examples

Find the matrix of the linear mapping. ${\displaystyle L(x,y,z)=(x+y+z)}$

The coefficient matrix for a system of linear differential equations of the form $y_1=Ay$
has the given eigenvalues and eigenspace bases. Find the general solution for the system
${\lambda }_1=3+i\Rightarrow \left\{\left[\begin{array}{c}2i\\ i\end{array}\right]\right\},{\lambda }_2=3-i\Rightarrow \left\{\left[\begin{array}{c}-2i\\ -i\end{array}\right]\right\}$

Consider the following two bases for ${\displaystyle R^3}$ :
$\alpha :=\{\left[\begin{array}{c}2\\ 1\\ 3\end{array}\right],\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right],\left[\begin{array}{c}3\\ 1\\ -1\end{array}\right]\}and\beta :=\{\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}-2\\ 3\\ 1\end{array}\right],\left[\begin{array}{c}2\\ 3\\ -1\end{array}\right]\}$ If $[x{]}_{\alpha }={\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]}_{\alpha }thenfind[x{]}_{\beta }$
(that is, express x in the ${\displaystyle \beta }$ coordinates).

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
${\displaystyle \left[\begin{array}{cccccc}1& -1& 0& -2& 0& 0\\ 0& 0& 1& 2& 0& 0\\ 0& 0& 0& 0& 1& 0\end{array}\right]}$

class council determined that its profit from the upcoming homecoming dance is directly related to the ticket price for the dance. Looking at past dances, the council determined that the profit pp can be modeled by the function $p(t)=-12t^2+480t+30$, where tt represents the price of each ticket. What should be the price of a ticket to the homecoming dance to maximize the council's profit? Price

Given f(x)=6x+5​, describe how the graph of g compares with the graph of f. g(x)=6(0.2x)+5
Select the correct choice below, and fill in the answer box to complete your choice.
A. The graph of​ g(x) is translated _ ​unit(s) to the left compared to the graph of​ f(x).
B. The graph of​ g(x) is translated _ ​unit(s) down compared to graph of​ f(x).
C. g(x) has a scale factor of _ compared to​ f(x). Because it scales the vertical​ direction, the graph is stretched vertically.
D. g(x) has a scale factor of _ compared to​ f(x). Because it scales the vertical​ direction, the graph is compressed vertically.
E. g(x) has a scale factor of _ compared to​ f(x). Because it scales the horizontal​ direction, the graph is stretched horizontally.
F. g(x) has a scale factor of _ compared to​ f(x). Because it scales the horizontal​ direction, the graph is compressed horizontally.
G. The graph of​ g(x) is translated _ ​unit(s) to the right compared to the graph of​ f(x).
H. The graph of​ g(x) is translated _ ​unit(s) up compared to graph of​ f(x).

Let ${\displaystyle \gamma =\{t^2-t+1,t+1,t^2+1\}\text{and}\beta =\{t^2+t+4,4t^2-3t+2,2t^2+3\}be\text{or}deredbasesf\text{or}{P}_2\left(R\right).}$ Find the change of coordinate matrix Q that changes $\beta \text{ coordinates into }\gamma -\text{ coordinates}$

To calculate: The intercepts on the coordinate axes of the straight line with the given equation ${\displaystyle 2y-4=3x}$

Give a full correct answer for given question 1- Let W be the set of all polynomials ${\displaystyle a+bt+c{t}^2\in P_2}$ such that ${\displaystyle a+b+c=0}$ Show that W is a subspace of ${\displaystyle P_2,}$ find a basis for W, and then find dim(W) 2 - Find two different bases of ${\displaystyle R^2}$ so that the coordinates of $b=\left[\begin{array}{c}5\\ 3\end{array}\right]$ are both (2,1) in the coordinate system defined by these two bases

The equivalent polar coordinates for the given rectangular coordinates.
A rectangular coordinate is given as (0, -3).

[Pic] Describe a combination of transformations.

A line passes through (9,3),(12,4), and (n,-5)
Find the value of n.

Let B and C be the following ordered bases of ${\displaystyle R^3:}$
$B=(\left[\begin{array}{c}1\\ 4\\ -\frac{4}{3}\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 8\end{array}\right],\left[\begin{array}{c}1\\ 1\\ -2\end{array}\right])$
$C=(\left[\begin{array}{c}1\\ 1\\ -2\end{array}\right],\left[\begin{array}{c}1\\ 4\\ -\frac{4}{3}\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 8\end{array}\right])$ Find the change of coordinate matrix I_{CB}

Consider the following vectors in ${\displaystyle R^4:}$ $v_1=\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right],v_2=\left[\begin{array}{c}0\\ 1\\ 1\\ 1\end{array}\right]v_3=\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right],v_4=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]$ d. If $x=\left[\begin{array}{c}23\\ 12\\ 10\\ 19\end{array}\right],find{\left\{x\right\}}_{B}e$. If $x_B=\left[\begin{array}{c}3\\ 1\\ -4\\ -4\end{array}\right]$, find x.

Find the Euclidean distance between u and v and the cosine of the angle between those vectors. State whether that angle is acute, obtuse, or ${\displaystyle {90}^{\circ }}$. u = (-1, -1, 8, 0), v = (5,6,1,4)

For any vectors u, v and w, show that the vectors u-v, v-w and w-u form a linearly dependent set.

The image of the point (2,1) under a translation is (5,-3). Find the coordinates of the image of the point (6,6) under the same translation.

Consider the linear transformation ${\displaystyle U:R^3\to R^3}$ defined by $U\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}z-y\\ z+y\\ 3z-x-y\end{array}\right)$ and the bases $ϵ=\{\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)\},\gamma =\{\left(\begin{array}{c}1-i\\ 1+i\\ 1\end{array}\right),\left(\begin{array}{c}-1\\ 1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)\}$, Compute the four coordinate matrices ${\displaystyle {\left[U\right]}_{ϵ}^{\gamma },{\left[U\right]}_{\gamma }^{\gamma },}$

Given the full and correct answer the two bases of $B1=\{\left(\begin{array}{c}2\\ 1\end{array}\right),\left(\begin{array}{c}3\\ 2\end{array}\right)\}$
$B_2=\{\left(\begin{array}{c}3\\ 1\end{array}\right),\left(\begin{array}{c}7\\ 2\end{array}\right)\}$
find the change of basis matrix from ${\displaystyle B_1\to B_2}$ and next use this matrix to covert the coordinate vector
${\overrightarrow{v}}_{B_1}=\left(\begin{array}{c}2\\ -1\end{array}\right)$ of v to its coodirnate vector
${\overrightarrow{v}}_{B_2}$

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)