A city planner wanted to place the new town library at site A. The mayor thought that it would be better at site B. What transformations were applied to the building at site A to relocate the building to site B? Did the mayor change the size or orientation of the library? [Pic]

Find the vectors T, N, and B at the given point.
$r(t)=<t^2,\frac{2}{3}{t}^3,t>$ and point $<4,-\frac{16}{3},-2>$

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR
Consider the points below
P(1,0,1) , Q(-2,1,4) , R(7,2,7)
a) Find a nonzero vector orthogonal to the plane through the points P,Q and R
b) Find the area of the triangle PQR

Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at ${\displaystyle (1,3,0),(-2,0,2),\text{and}(-1,3,-1)}$.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A system of linear equations in three variables, x, y, and z cannot contain an equation in the form $y=mx+b$.

Let $v\in T_2(V)$ be a bilinear form over finite space V. Let T be a Linear transformation $V\to V$. We define: $v_T(x,y)=v(T(x),y)$
Assuming $v$ is nondegenerate, let us have another bilinear form $\xi \in T_2(V)$. Prove that there exists exactly one transformation $T$ so $\xi =v_T$.

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 $c>0$

Let u and v be distinct vectors of a vector space V. Show that if {u, v} is a basis for V and a and b are nonzero scalars, then both {u+v, au} and {au, bv} are also bases for V.