Discoverand Master Vectors Solutions

Find a basis for the space spanned by the given vectors, ${\displaystyle v_1\dots ,v_5}$
$$$\left[\begin{array}{ccccc}1& -2& 6& 5& 0\\ 0& 1& -1& -3& 3\\ 0& -1& 2& 3& -1\\ 1& 1& -1& -4& 1\end{array}\right]\sim \left[\begin{array}{ccccc}1& 0& 0& -1& -2\\ 0& 1& 0& -3& 5\\ 0& 0& 1& 0& 2\\ 0& 0& 0& 0& 0\end{array}\right]$

List five vectors in Span ${\displaystyle \{v_1,v_2\}}$. For each vector, show the weights on ${\displaystyle v_1}$ and ${\displaystyle v_2}$ used to generate the vector and list three entries of the vector. Do not make a seketch.
$v_1=\left[\begin{array}{c}7\\ 1\\ -6\end{array}\right],v_2=\left[\begin{array}{c}-5\\ 3\\ 0\end{array}\right]$

if $x,y$ belong to $R^p$, than is it true that the relation ${\displaystyle \Vert x+y\Vert =\Vert x\Vert +\Vert y\Vert }$ holds if and only if ${\displaystyle x=cy\text{or}y=cx}$ with $c>0$

Find the solid's first octant's volume, which is enclosed by the cylinder ${\displaystyle z=16-x^2}$ and the plane y=5

Solve the rate of change of f at P in the direction of the vector u. ${\displaystyle f(x,y,z)=y^2{e}^{x}yz,P(0,1,-1),u=<\frac{3}{13},\frac{4}{13},\frac{12}{13}>}$

Find the minimum and maximum values of ${\displaystyle y=\sqrt{6}\theta -\sqrt{3}\sec \theta }$
on the interval $[0,\frac{\text{pi}}{\text{3}}]$

4.(a) Sketch the region of solutions of the following systems of inequalities:
${\displaystyle 2?+3?\le 12}$
${\displaystyle -?+2?<4}$
${\displaystyle ?\ge 0}$
${\displaystyle ?\ge 0}$
b. List the coordinates of all of the corner points:
c Which corner point would maximize the equation ${\displaystyle ?=5?+3??}$