Orthogonal functions: Difference between revisions
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==Introduction== |
==Introduction== |
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In this article we will examine another viewpoint for functions than that traditionally taken. Normally we think of a function, f(t), as a complicated entity in a f(), in a simple environment (one dimension, or along the t axis). Now we want to think of a function as a vector or point (a simple thing) in a very complicated environment (possibly an infinite dimensional space). |
In this article we will examine another viewpoint for functions than that traditionally taken. Normally we think of a function, f(t), as a complicated entity in a f(), in a simple environment (one dimension, or along the t axis). Now we want to think of a function as a vector or point (a simple thing) in a very complicated environment (possibly an infinite dimensional space). |
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==Vectors== |
==Vectors== |
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Recall that vectors consist of an ordered set of numbers. Often the numbers are Real numbers, but we shall allow them to be from the Complex numbers for our purposes. The numbers represent the amount of the vector in the direction denoted by the position of the number in the list. Each position in the list is associated with a direction. For example, the vector |
Recall that vectors consist of an ordered set of numbers. Often the numbers are Real numbers, but we shall allow them to be from the Complex numbers for our purposes. The numbers represent the amount of the vector in the direction denoted by the position of the number in the list. Each position in the list is associated with a direction. For example, the vector |
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===Notation=== |
===Notation=== |
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We don't have to use x, y, and z as the direction names; we can use numbers, like 1, 2, and 3 instead. The advantage of this is that it leads to more compact notation, and extends to more than three dimensions much better. For example we could say <math> v_2 = 4 </math> instead of <math> v_y = 4 </math>. Instead of writing <math> \vec \bold v = <1, 4, 3> </math> we can write <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math> where the <math>\hat \bold a_k </math> denotes a unit vector (or basis vector) in the kth direction. The idea of basis vectors was implicit in the notation <math> \vec \bold v = <1, 4, 3> </math>. |
We don't have to use x, y, and z as the direction names; we can use numbers, like 1, 2, and 3 instead. The advantage of this is that it leads to more compact notation, and extends to more than three dimensions much better. For example we could say <math> v_2 = 4 </math> instead of <math> v_y = 4 </math>. Instead of writing <math> \vec \bold v = <1, 4, 3> </math> we can write <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math> where the <math>\hat \bold a_k </math> denotes a unit vector (or basis vector) in the kth direction. The idea of basis vectors was implicit in the notation <math> \vec \bold v = <1, 4, 3> </math>. |
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| ⚫ | We may think of the number of the direction, <math> k </math>, as the independent variable of a vector and the component in that direction, <math> v_k </math> as the dependent variable of the vector <math> \vec \bold v </math> in a similar way to the way we think of t as the independent variable of a function f(), where f(t) is the dependent variable of f. Probably the biggest difference here is that t often takes on real values from <math> - \infty </math> to <math> \infty </math>, and <math> k \in {1, 2, 3} </math>. |
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===Changing Basis Sets=== |
===Changing Basis Sets=== |
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Sometimes in our studies we find it useful to change basis sets. For example, when solving a physics problem with cylindrical symmetry, it is often easier to use cylindrical coordinates, and the basis vectors that go with that system, rather than the more usual Cartesian coordinates and basis vectors. Let us remind ourselves how we do the transformation from a one coordinate system to another. (Put more here.) |
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==Functions and Vectors, an Analogy== |
==Functions and Vectors, an Analogy== |
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| ⚫ | We may think of the number of the direction, <math> k </math>, as the independent variable of a vector and the component in that direction, <math> v_k </math> as the dependent variable of the vector <math> \vec \bold v </math> in a similar way to the way we think of t as the independent variable of a function f(), where f(t) is the dependent variable of f. Probably the biggest difference here is that t often takes on real values from <math> - \infty </math> to <math> \infty </math>, and <math> k \in {1, 2, 3} </math>. |
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Revision as of 18:35, 13 September 2004
Introduction
In this article we will examine another viewpoint for functions than that traditionally taken. Normally we think of a function, f(t), as a complicated entity in a f(), in a simple environment (one dimension, or along the t axis). Now we want to think of a function as a vector or point (a simple thing) in a very complicated environment (possibly an infinite dimensional space).
Vectors
Recall that vectors consist of an ordered set of numbers. Often the numbers are Real numbers, but we shall allow them to be from the Complex numbers for our purposes. The numbers represent the amount of the vector in the direction denoted by the position of the number in the list. Each position in the list is associated with a direction. For example, the vector means that the vector is one unit in the first direction (often the x direction), four units in the second direction (often the y direction), and three units in the last direction (often the z direction). We say the component of in the second direction is 4. This is often written as .
Notation
We don't have to use x, y, and z as the direction names; we can use numbers, like 1, 2, and 3 instead. The advantage of this is that it leads to more compact notation, and extends to more than three dimensions much better. For example we could say instead of . Instead of writing we can write where the denotes a unit vector (or basis vector) in the kth direction. The idea of basis vectors was implicit in the notation .
Changing Basis Sets
Sometimes in our studies we find it useful to change basis sets. For example, when solving a physics problem with cylindrical symmetry, it is often easier to use cylindrical coordinates, and the basis vectors that go with that system, rather than the more usual Cartesian coordinates and basis vectors. Let us remind ourselves how we do the transformation from a one coordinate system to another. (Put more here.)
Functions and Vectors, an Analogy
Independent and Dependent Variables
We may think of the number of the direction, , as the independent variable of a vector and the component in that direction, as the dependent variable of the vector in a similar way to the way we think of t as the independent variable of a function f(), where f(t) is the dependent variable of f. Probably the biggest difference here is that t often takes on real values from to , and .
