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Vector Equations
Equation of linear combination of vectors with unknown coefficients. Essentially defines a linear system. For example, \(\cos(x) + \sin(y) = 0\) is NOT linear.
There three ways to think of them:
- System of linear equations
\[
\begin{matrix}
2x_1 + 3x_2-3x_3=7\\
x_1-x_2-3x_3=5
\end{matrix}
\]
- Augmented Matrix: Representing the equation as a matrix and allowing elimination operations easily when done by hand.
- Algorithms: Gaussian Elimination and Row Reduction.
- To solve: All of them are methods to get augmented matrix in reduced row echelon form.
- Reduced: Where all pivots are equal to 1, and are the only non-zero entry in the column.
\[
\left[
\begin{array}{ccc|c}
2 & 3 & -2 & 7 \\
1 & -1 & -3 & 5
\end{array}
\right]
\]
- Vector Equation
\[
x_1 \begin{pmatrix}
2\\1
\end{pmatrix} + x_2 \begin{pmatrix}
3\\-1
\end{pmatrix} + x_3 \begin{pmatrix}
-2\\-3
\end{pmatrix} = \begin{pmatrix}
7\\5
\end{pmatrix}
\]
A system of equation is consistent if it has a solution.
If there are no solutions, it is inconsistent.