Simplifying Linear Algebra: Techniques, Solutions, and Examples

Find the best approximation to z by vectors of the form ${\displaystyle c_1{v}_1+c_2{v}_2}$
$z=\left[\begin{array}{c}3\\ -7\\ 2\\ 3\end{array}\right],v_1=\left[\begin{array}{c}2\\ -1\\ -3\\ 1\end{array}\right]v_2=\left[\begin{array}{c}1\\ 1\\ 0\\ -1\end{array}\right]$

The row echelon form of a system of linear equations is given.
(a)Writethesystemofequationscorrespondingtothegivenmatrix.Usex,y.or x,y,z.or $x_1,x_2,x_3,x_4$ asvariables.
(b)Determinewhetherthesystemisconsistentorinconsistent.Ifitisconsistent,givethesolution.

$\left[\begin{array}{cccc}1& 2& |& 5\\ 0& 1& |& -1\end{array}\right]$

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.

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

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.

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

(1) find the projection of u onto v and (2) find the vector component of u orthogonal to v. u = ⟨6, 7⟩, v = ⟨1,4⟩

We need to find the volume of the parallelepiped with only one vertex at the origin and conterminous vertices at ${\displaystyle (1,3,0),(-2,0,2),\text{and}(-1,3,-1)}$.

Let S be the parallelogram determined by the vectors
$b_1=\left[\begin{array}{c}-2\\ 3\end{array}\right]$
and
$b_2=\left[\begin{array}{c}-2\\ 5\end{array}\right]$
and let
$A=\left[\begin{array}{cc}6& -3\\ -3& 2\end{array}\right]$
Compute the area S under the mapping ${\displaystyle x\mapsto Ax}$

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; or x, y, z;x,y,z; or ${\displaystyle x_1,x_2,x_3,x_4}$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
$\left[\begin{array}{cccccc}1& 0& 0& 0& |& 1\\ 0& 1& 0& 0& |& 2\\ 0& 0& 1& 2& |& 3\end{array}\right]$

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 $x_1,x_2,\dots x,\dots$ from left to right.

$\left[\begin{array}{cccc}1& 0& 0& -3\\ 0& 1& 0& 0\\ 0& 0& 1& 7\end{array}\right]$

Find an orthogonal basis for the column space of each matrix.
$\left[\begin{array}{ccc}3& -5& 1\\ 1& 1& 1\\ -1& 5& -2\\ 3& -7& -8\end{array}\right]$

Consider the linear system
${\overrightarrow{y}}^{\prime }=\left[\begin{array}{cc}-3& -2\\ 6& 4\end{array}\right]\overrightarrow{y}$
a) Find the eigenvalues and eigenvectors for the coefficient matrix
${\lambda }_{1}1,{\overrightarrow{v}}_1=\left[\begin{array}{c}-1\\ 2\end{array}\right]$ and ${\lambda }_2=0,{\overrightarrow{v}}_2=\left[\begin{array}{c}-2\\ 3\end{array}\right]$
b) For each eigenpair in the previos part, form a solution of ${\displaystyle {\overrightarrow{y}}^{\prime }=A\overrightarrow{y}}$. Use t as the independent variable in your answers.
${\overrightarrow{y}}_1(t)=\left[\begin{array}{cc}& \\ & \end{array}\right]$ and ${\overrightarrow{y}}_2(t)=\left[\begin{array}{c}-2\\ 3\end{array}\right]$
c) Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solution? No, it is not a fundamental set.