Discoverand Master Vectors Solutions

m Find m $Y=(6x+44)$
$Z=(-10x+65)$

Let ${\displaystyle B=\{v1,v2,\dots ,vm\}}$ be a basis for $R^m$. Suppose kvm is a linear combination of $v1,v2,\cdots ,vm-1$ for some scalar k. What can be said about the possible value(s) of k?

Given the vector $r(t)=\cos T,\sin T,\ln (\cos T)$ and point (1, 0, 0) find vectors T, N and B at that point.

Vector T is the unit tangent vector, so the derivative r(t) is needed.

Vector N is the normal unit vector, and the equation for it uses the derivative of T(t). $$

The B vector is the binormal vector, which is a crossproduct of T and N.

It is given: ${\displaystyle \parallel m\parallel =4,\parallel n\parallel =\sqrt{2},⟨m,n⟩=135}$ find the norm of the vector $m+3n$.

Let S be a subset of an F-vector space V. Show that V is a subspace of Span(S).

Write the vector v in the form ai + bj, given its magnitude ∥v∥∥v∥ and the angle $\alpha$ it makes with the positive x-axis. $\parallel v\parallel =8,\alpha =45\ast \parallel v\parallel =8,\alpha =45$

Consider ${\displaystyle V=\cos \left(x\right),\sin \left(x\right)}$ a subspace of the vector space of continuous functions and a linear transformation ${\displaystyle T:V\to V}$ where ${\displaystyle T\left(f\right)=f\left(0\right)\times \cos \left(x\right)-f\left(\pi 2\right)\times \sin \left(x\right).}$

Find the matrix of T with respect to the basis ${\displaystyle \cos \left(x\right)+\sin \left(x\right),\cos \left(x\right)-\sin \left(x\right)}$ and determine if T is an isomorphism.