Simplifying Linear Algebra: Techniques, Solutions, and Examples

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]$

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

Let A be an ${\displaystyle m\times n}$ matrix, and ${\displaystyle C=AB}$.
Show that:
a) $Null(B)$ is a subspace of $Null(C)$.
b) $Null(C{)}^{\mathrm{\perp }}$ is a subspace of $Null(B{)}^{\mathrm{\perp }}$ and, consequently, ${\displaystyle Col\left(CT\right)}$ is a subspace of ${\displaystyle Col\left(BT\right)}$.

T must be a linear transformation, we assume. Can u find the T standard matrix.${\displaystyle T:{\mathbb{R}}^2\to {\mathbb{R}}^4,T\left(e_1\right)=(3,1,3,1)\text{and}T\left(e_2\right)=(-5,2,0,0),where{e}_1=(1,0)\text{and}{e}_2=(0,1)}$.

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent. $\left[\begin{array}{ccccc}1& 0& -1& |& 1\\ 0& 1& 2& |& 1\end{array}\right]$

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
${\lambda }_1=2\Rightarrow \left\{\left[\begin{array}{c}4\\ 3\\ 1\end{array}\right]\right\},{\lambda }_2=-2\Rightarrow \{\left[\begin{array}{c}1\\ 2\\ 0\end{array}\right],\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]\}$

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named x1,x2,…x,… form left to right. $\left[\begin{array}{cccc}1& -3& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$