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43 lines
1.4 KiB
Markdown
43 lines
1.4 KiB
Markdown
#math/linear_algebra
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### matrix representation
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#card
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a system can be represented by a matrix. each column represents the coefficients of each variable involved, the rows represent each equation in the system. Optionally, an augmented matrix includes a bar followed by the constant (but that's not there when the constant is 0)
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For a given system of equations
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$$
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\begin{aligned}
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x_{1}+3 x_{2}+2 x_{3}+3 x_{4} &=-4 \\
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x_{2}-2 x_{3}-2 x_{4} &=3 \\
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-x_{1}-3 x_{2}+2 x_{3}+x_{4} &=4
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\end{aligned}
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$$
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we can express it as a matrix of coefficients
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$$
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\left[\begin{array}{rrrr}
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1 & 3 & 2 & 3 \\
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0 & 1 & -2 & -2 \\
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-1 & -3 & 2 & 1
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\end{array}\right]
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$$
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or as an augmented matrix
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$$
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\left[\begin{array}{rrrr|r}
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1 & 3 & 2 & 3 & -4 \\
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0 & 1 & -2 & -2 & 3 \\
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-1 & -3 & 2 & 1 &
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\end{array}\right]
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$$
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![[Pasted image 20220502151226.png]]
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^1651526044974
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#### solve
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You can do whatever you want with these elementary operations:
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![[Pasted image 20220502232912.png]]
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You wanna use those operations to reduce the matrix to row echelon form like so:
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##### (reduced) row echelon form
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![[Pasted image 20220502233033.png]]
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these are often solutions to the system of equations. If a row and column intersection has just a one, then the little margin on the right signifies the solution for that column's corresponding variable.
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##### the algorithm
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the procedure to do so is as follows:
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![[Pasted image 20220505172055.png]] |