Simplifying Linear Algebra: Techniques, Solutions, and Examples

T must be a linear transformation, we assume. Identify the T standard matrix. 𝟚𝟚${\displaystyle T:{\mathbb{R}}^{\mathbb{2}}\to {\mathbb{R}}^{\mathbb{2}}}$ is a vertical shear transformation that maps $e_{1}into{e}_1-2e_2$ but leaves the vector ${\displaystyle e_2}$ unchanged.

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. $P(3,0,1),Q(-1,2,5),R(5,1,-1),S(0,4,2)$

Reduce the system of linear equations to upper triangular form and solve.
$\{\begin{array}{l}2x-y=3\\ x-3y=7\end{array}$

The row echelon form of a system of linear equations is given.
(a)Writethesystemofequationscorrespondingtothegivenmatrix.Use x,y.or x,y,z.or $x_1,x_2,x_3,x_4$ asvariables.
(b)Determinewhetherthesystemisconsistentorinconsistent.Ifitisconsistent,givethesolution. $\left[\begin{array}{cccc}1& -3& |& -4\\ 0& 1& |& 0\end{array}\right]$

What is the area of the parallelogram whose vertices are listed? (0,0), (5,2), (6,4), (11,6)

Let W be the subspace spanned by the given vectors. Find a basis for
$w_1=\left[\begin{array}{c}1\\ -1\\ 3\\ -2\end{array}\right],w_2=\left[\begin{array}{c}0\\ 1\\ -2\\ 1\end{array}\right]$

Use the cross product to find the sine of the angle between the vectors ${\displaystyle u=(2,3,-6)}$ and $v=(2,3,6)$.

Find ${\displaystyle \overrightarrow{a}+\overrightarrow{b},2\overrightarrow{a}+3\overrightarrow{b},\left|\overrightarrow{a}\right|}$ and $|\overrightarrow{a}-\overrightarrow{b}|$.
${\displaystyle \overrightarrow{a}=<5,-12>,\overrightarrow{b}=<-3,-6>}$

The work W done by a constant force F in moving an object from a point A in space to a point B in space is defined as $W=F\cdot \overrightarrow{AB}$. Find the work done by a force of 3 newtons acting in the direction ${\displaystyle 2i+j+2k}$ in moving an object 2 meters from ${\displaystyle (0,0,0)}$ to ${\displaystyle (0,2,0)}$.

Without calculation, find one eigenvalue and two linearly independent eigenvectors of
$A=\left[\begin{array}{ccc}5& 5& 5\\ 5& 5& 5\\ 5& 5& 5\end{array}\right]$
Justify your answer.

Find the cross product $a\times b$ and verify that it is orthogonal to both a and b.
$a=(2,3,0),b=(1,0,5)$

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 $x_1,x_2,\dots$from left to right. $\left[\begin{array}{ccccc}1& 0& 5& 3& 2\\ 0& 1& -2& 4& -7\\ 0& 0& 1& 1& 3\end{array}\right]$

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.

$\left[\begin{array}{ccccc}1& 0& -2& |& 6\\ 0& 1& 3& |& 1\end{array}\right]$