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This tutorial assumes you know what either the dot product or the [[Inner product | inner product]] and what [[Orthogonal|orthonormal]] [[Basis| bases]] are. A good review can be found '''[[Inner_Products_MA265F11Walther|HERE]]'''. First we will look at the 3 dimensional case over the real numbers like in [[MA265]] , and then we will extend our understanding to the n-dimensional case over F like in [[MA351]] or '''MA353'''. We will also take a look at these problems from a computational standpoint, as is done in '''CS314''', to give light to the purpose behind QR-factorization. | This tutorial assumes you know what either the dot product or the [[Inner product | inner product]] and what [[Orthogonal|orthonormal]] [[Basis| bases]] are. A good review can be found '''[[Inner_Products_MA265F11Walther|HERE]]'''. First we will look at the 3 dimensional case over the real numbers like in [[MA265]] , and then we will extend our understanding to the n-dimensional case over F like in [[MA351]] or '''MA353'''. We will also take a look at these problems from a computational standpoint, as is done in '''CS314''', to give light to the purpose behind QR-factorization. | ||
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+ | ==Gram-Schmidt== | ||
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+ | ===Projections=== | ||
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+ | Let's say we have two vectors <math>\vec{v_1} = \{ 3, 2, 0\}\quad and\quad\vec{v_2} = \{1, 2, 0\}</math>. | ||
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+ | <gallery> | ||
+ | File:projection.jpg|Caption1 | ||
+ | </gallery> | ||
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+ | The Gram-Schmidt process is an algorithm by which we can orthogonalize a basis. The idea is the following. | ||
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== Intuition == | == Intuition == | ||
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The first thing I'm going to do is plot my data on a graph and sketch a line that I think is a good estimate for my data. It will look something like this. | The first thing I'm going to do is plot my data on a graph and sketch a line that I think is a good estimate for my data. It will look something like this. | ||
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+ | ==Sources== | ||
http://jccc-mpg.wikidot.com/vector-projection | http://jccc-mpg.wikidot.com/vector-projection |
Latest revision as of 21:37, 12 November 2017
Prerequisite Knowledge
This tutorial assumes you know what either the dot product or the inner product and what orthonormal bases are. A good review can be found HERE. First we will look at the 3 dimensional case over the real numbers like in MA265 , and then we will extend our understanding to the n-dimensional case over F like in MA351 or MA353. We will also take a look at these problems from a computational standpoint, as is done in CS314, to give light to the purpose behind QR-factorization.
Gram-Schmidt
Projections
Let's say we have two vectors $ \vec{v_1} = \{ 3, 2, 0\}\quad and\quad\vec{v_2} = \{1, 2, 0\} $.
The Gram-Schmidt process is an algorithm by which we can orthogonalize a basis. The idea is the following.
Intuition
Let's say I've heard that the relationship between a person's height and their wingspan is approximately a 1:1 ratio. I'd like to test this theory and also come up with an equation that can estimate wingspan given a height. To get started, I've collected some sample data from a few of my friends.
Height (in.) | Wingspan (in.) |
67.75 | 67.25 |
64.25 | 64.50 |
75.75 | 73.50 |
62.25 | 61.75 |
62.75 | 60.75 |
66.75 | 66.25 |
The first thing I'm going to do is plot my data on a graph and sketch a line that I think is a good estimate for my data. It will look something like this.