College math questions and answers

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)

The type of conic sections for the nondegenerate equations given below.

a) ${\displaystyle 6x^2+3x+10y=10y^2+8}$

b) ${\displaystyle 3x^2+18xy=5x+2y+9}$

c) ${\displaystyle 4x^2+8x-5=-y^2+6y+3}$

Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by
$F(s)={\int }_0^{\mathrm{\infty }}{e}^{-st}f(t)dt$
where we assume s is a positive real number. For example, to find the Laplace transform of $f(t)=e^{-t}$, the following improper integral is evaluated using integration by parts:
$F(s)={\int }_0^{\mathrm{\infty }}{e}^{-st}{e}^{-t}dt={\int }_0^{\mathrm{\infty }}{e}^{-(s+1)t}dt=\frac{1}{(s+1)}$
Verify the following Laplace transforms, where u is a real number.
$f(t)=1\to F(s)=\frac{1}{s}$