16 lines
1.2 KiB
Markdown
16 lines
1.2 KiB
Markdown
#uw/notes #uw/class/math208
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- Not all systems of linear equations will be in echelon form and easily solvable
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- There are three **elementary operations** that can be used to create a new system that is equivalent to the old one
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- Interchange the position of two equations
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- Multiply an equation by a nonzero constant
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- Add a multiple of one equation to another
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- $\sim$ is used to indicate the transformation between equivalent linear systems
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- matrices can be used to simplify when working with systems of linear equations
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- matrices with all constant terms of a linear system of equations are called a **augmented matrix**
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- $$\begin{gathered} \text{Linear System} \\ a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} = b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3} = b_{2} \\ a_{31}x_{1} + a_{32}x_{2} + a_{33}x_{3} = b_{3} \end{gathered} \quad \sim \quad \begin{gathered} \text{Augmented Matrix} \\ \begin{bmatrix}
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a_{11} & a_{12} & a_{13} & b_{1} \\ a_{21} & a_{22} & a_{23} & b_{2} \\ a_{31} & a_{32} & a_{33} & b_{3}
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\end{bmatrix} \end{gathered}$$
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- Same elementary operations can be used with augmented matrices, now it is with rows instead of equations.
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- gaussian elimination
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- this is converting a matrix to **echelon form** (or row echel) |