Simplifying Linear Algebra: Techniques, Solutions, and Examples

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.
${\displaystyle \lambda 1=-1\to \left\{\begin{array}{cc}1& 1\end{array}\right\},\lambda 2=2\to \left\{\begin{array}{cc}1& -1\end{array}\right\}}$

if x, y belong to $R^p$, than is it true that the relation norm $(x+y)=norm(x)+norm(y)$ holds if and only if $x=cyory=cxwithc>0$

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. $\left[\begin{array}{ccccc}1& 0& -1& |& 1\\ 0& 1& 2& |& 1\end{array}\right]$

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 $x_1,x_2\dots$ from left to right. $[1,2,0,2,-1,3]$

For the matrix A below, find a nonzero vector in the null space of A and a nonzero vector in the column space of A
$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$ 
Find a vector in the null space of A that is not the zero vector 
$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

Find $|u\cdot v|$ and determine whether $u\cdot v$ 
is pointed at the screen or away from it.
$|u\cdot v|=??$

Each of the matrices is the final matrix form for a system of two linear equations in the variables x1 and x2. Write the solution of the system.
$\left[\begin{array}{cccc}1& -2& |& 7\\ 0& 0& |& -9\end{array}\right]$

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x,y;x,y;orx,y,z;x,y,z;or{x}_1,x_2,x_3,x_4$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.

$\left[\begin{array}{ccccc}1& 0& 2& |& -1\\ 0& 1& -4& |& -2\\ 0& 0& 0& |& -0\end{array}\right]$

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution.

$\left[\begin{array}{cccc}1& -2& 0& -4\\ 0& 0& 1& 3\\ 0& 0& 0& 0\end{array}\right]$

Assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance ${\displaystyle {\sigma }^2}$ and (b) the population standard deviation ${\displaystyle \sigma }$. Interpret the results. The final exam scores of 24 randomly selected students in a statistics class are shown in the table at the left. Use a 95% level of confidence.

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given row echelon form. Solve the system by back substitution. Assume that the variables are named $x_1,x_2,$from left to right. $\left[\begin{array}{ccccc}1& 0& 5& 3& 2\\ 0& 1& -2& 4& -7\\ 0& 0& 1& 1& 3\end{array}\right]$

At the specified position, determine the vectors T, N, and B.
$r(t)=<t^2,\frac{2}{3}{t}^3,t>$ and point $<4,-\frac{16}{3},-2>$

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution.
Write the solution in vector form.

$\left[\begin{array}{ccccc}1& 0& 0& -3& 0\\ 0& 1& 0& -4& 0\\ 0& 0& 1& 5& 0\end{array}\right]$