Simplifying Linear Algebra: Techniques, Solutions, and Examples

Systems of inequalities graph the solutions set of the system if inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $\{\begin{array}{l}x+y\le 4\\ y\ge x\end{array}$

To solve:
the system of equation ${\displaystyle 2x+3y=5\text{and}6x+9y=15}$.

The coordinate vector of ${\displaystyle [\mathbf{p}\left(t\right)=6+3t-t^2]}$ relative to the basis ${\displaystyle \left\{\mathcal{B}\right\}=\{1+t,1+t^2,t+t^2\}}$

Find the values of k for which A is not invertible. Enter all values exactly in fractional form.
$A=[1,k,0,9,1,9,0,k,1]$

Use the weighted Euclidean inner product on $R^2$ $‹u, v› = 99u\_1v\_1 + 5u\_2v\_2$ where $u=(u_1,u_2)$ and $v=(v_1,v_2)$, to find $\left|\right|w\left|\right|$, where $w = (- 1, 3)$.

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.