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Section 13.1 Linear Algebra

Lines and planes in two and three dimensional spaces

Equation of a line in two dimensional space can be given by

\begin{gather*} ax+by=c \end{gather*}

and equation of a plane in three dimentional space can be given as

\begin{gather*} ax + by + cz = d\text{.} \end{gather*}

These equations are called linear because the power of the variables are restricted to be one, and because they represent geometric shapes which are Straight, and in three dimensions are flat.

Here are few examples of equations of lines in two dimentional space:
\begin{gather*} {\color{blue}{x=2, y =3, 2x-3y = 4 }} \end{gather*}
Here are few examples of equations of planes in three dimentional space:
\begin{gather*} {\color{magenta}{x=2, y =3, 3x - 2z = 6, 2x-3y + 5z = 4 }}. \end{gather*}
The following equations are not Linear.
\begin{gather*} {\color{purple}{ x^2 - x =2, \sqrt { yx - z} =3, 3x \sin y - 2z = 6, 2x^2-3y^2 + 5z^2 = 4 }} \end{gather*}

What if we have Non-linear Equations?Hint

Non-Linear eauations can be linearized before applying linear algebra.