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| == Dot product ==
| | #REDIRECT [[Linear algebra formalism]] |
| A vector is a list of numbers, written like so: <math> \begin{pmatrix}
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| -2\\
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| 0\\
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| 1
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| \end{pmatrix} </math>.
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| The dot product is a way to combine two vectors to get out a single number.
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| Say we want to take the dot product of the vectors <math> \begin{pmatrix}12\\19\\28\end{pmatrix} </math> and <math> \begin{pmatrix}-2\\0\\1\end{pmatrix} </math>.
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| To do so, follow these steps:
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| * Write the vectors separated by a dot to denote the dot product: <math>
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| \begin{pmatrix}
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| 12\\
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| 19\\
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| 28\\
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| \end{pmatrix}
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| \cdot
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| \begin{pmatrix}
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| -2\\
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| 0\\
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| 1
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| \end{pmatrix}
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| </math>
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| **This may also be notated <math> \langle 12, 19, 28 \vert -2, 0, 1\rangle </math>; from this derives the notation for vals and monzos.
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| * Multiply the corresponding elements, and add the results together: <math> \left(12\cdot-2\right)+\left(19\cdot0\right)+\left(28\cdot1\right) = -24 + 0 + 28 = 4 </math>
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| == Multiply matrix by vector ==
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| A matrix is a grid of numbers, written like so:
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| <math>
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| \begin{bmatrix}
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| 1 & 0 & -4\\
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| 0 & 1 & 4
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| \end{bmatrix}
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| </math>
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| This matrix can be thought of as a "function" that you apply to a vector to get out another vector. This matrix has 3 columns, meaning the vector it takes as an "input" will have 3 elements, and it has 2 rows, meaning the vector you get out will have 2 elements. So, this is a "function" down from 3-dimensional space to 2-dimensional space.
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| We write this "application" of a matrix like so:
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| <math>
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| \begin{bmatrix}
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| 1 & 0 & -4\\
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| 0 & 1 & 4
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| \end{bmatrix}
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| \begin{bmatrix}
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| -2\\
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| 0\\
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| 1
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| \end{bmatrix}
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| </math>
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| where the second object is the vector.
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| To write the first element of our output, we take the dot product of the first row of our matrix with our vector:
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| <math>\begin{pmatrix}
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| 1\\
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| 0\\
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| -4\\
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| \end{pmatrix}
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| \cdot
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| \begin{pmatrix}
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| -2\\
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| 0\\
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| 1
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| \end{pmatrix}
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| = \left(1\cdot-2\right)+\left(0\cdot0\right)+\left(-4\cdot1\right) = -2 + 0 + -4 = -6 </math>
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| We do the same thing for the second element of our output, computing <math>
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| \begin{pmatrix}
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| 0\\
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| 1\\
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| 4\\
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| \end{pmatrix}
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| \cdot
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| \begin{pmatrix}
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| -2\\
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| 0\\
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| 1
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| \end{pmatrix}
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| = 4
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| </math>.
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| Thus, our output is <math> \begin{bmatrix}
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| -6\\
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| 4\\
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| \end{bmatrix}</math>
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| .
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| == Multiply matrix by matrix ==
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| A matrix can act on another matrix, as well. In this case, the matrix on the right can simply be treated as several vectors next to each other.
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| <math>
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| \begin{bmatrix}
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| 1 & 0 & -4\\
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| 0 & 1 & 4
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| \end{bmatrix}
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| \begin{bmatrix}
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| 1 & -1 & -2\\
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| 0 & 1 & 0\\
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| 0 & 0 & 1
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & -1 & -6\\
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| 0 & 1 & 4
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| \end{bmatrix}
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| </math>
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