Difference between revisions of "Linear Algebra"

Revision as of 09:36, 25 August 2008

Basics

Let's first make some definitions:

Vector: A one-dimensional collection of numbers. Examples:; $a=\left( \begin{array}{cccc} 1 & 2 & 3 & 4 \end{array} \right), \quad b=\left( \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right)$

Matrix: A two-dimensional collection of numbers. Examples:; $A = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right], \quad B = \left[ \begin{array}{cc} 9 & 6 \\ 2 & 3 \end{array}\right]$

Array: An n-dimensional collection of numbers. A vector is a 1-dimensional array while a matrix is a 2-dimensional array.

Transpose: An operation that involves interchanging the rows and columns of an array. It is indicated by a $^\mathsf{T}$ superscript. For example:; $a=\left( \begin{array}{cccc} 1 & 2 & 3 & 4 \end{array} \right) \; \Rightarrow \; a^\mathsf{T} = \left( \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right)$; $A = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right] \; \Rightarrow \; A^\mathsf{T} = \left[ \begin{array}{cc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array} \right]$

Vector Magnitude, $\left|\vec{a}\right|$: A measure of the length of a vector, $\left|\vec{a}\right| = \sqrt{ \sum_{i=1}^{n} a_i^2 }$; Example: $\vec{a}=\left( \begin{array}{ccc} 2 & 8 & 3 \end{array} \right) \; \Rightarrow \; \left|\vec{a}\right| = \sqrt{ 2^2 + 8^2 + 3^2 }$; Example: $\vec{b}=\left( \begin{array}{cccc} b_1 & b_2 & b_3 & b_4\end{array} \right) \; \Rightarrow \; \left|\vec{b}\right| = \sqrt{ b_1^2 + b_2^2 + b_3^2 +b_4^2 }$

Matrix & Vector Algebra

There are many common operations involving matrices and vectors including:

Vector dot product
Vector cross product
Matrix-vector product
Matrix-matrix product

These are each discussed in the following sub-sections.

Vector Dot Product

The dot product of two vectors produces a scalar. Physically, the dot product of a and b represents the projection of a onto b.

Given two vectors a and b, their dot product is formed as

c = \vec{a} \cdot \vec{b} = \sum_{i=1}^n a_i b_i,

where $a_i$ and $b_i$ are the components in vectors a and b respectively. This is most useful when we know the components of the vectors a and b.

Occasionally we know the magnitude of the two vectors and the angle between them. In this case, we can calculate the dot product as

|c| = |a|\,|b|\,\cos(\theta),

where θ is the angle between the vectors a and b.

For more information on the dot product: [1]

Thought examples

Consider the cartoon shown to the right.

At noon, when the sun is directly overhead, a rocket is launched directly toward the sun at 1000 mph. How fast is its shadow moving?

Define the rocket's velocity as

\vec{R}

.

Define the unit normal on the ground (i.e. the direction of the rocket's shadow on the ground) as

\vec{s}

, where

|\vec{s}| = 1

.

Intuition tells us that if the it is moving directly toward the sun, then its shadow does not appear to move at all. : The dot product

\vec{R}\cdot\vec{s} = |\vec{R}| \, |\vec{s}| \cos(\theta) = 0

since cos(90°)=0.

Consider the same rocket going parallel to the earth's surface. How fast is its shadow moving?

If the rocket is going parallel to the earth's surface, our intuition tells us that is shadow is moving at the same speed. This is confirmed by the mathematics, since the angle between the rocket's path and the ground is 0. Therefore,

\vec{R}\cdot\vec{s} = |\vec{R}| \, |\vec{s}| \cos(\theta) = 0

since cos(0°)=1.

What if the rocket were going at a 45° angle?

Our intuition tells us that the shadow will appear to move, but it will not be moving as fast as the rocket is moving. Mathematically, we have

v_s = \vec{R}\cdot\vec{s} = |\vec{R}| |\vec{s}| \cos(45^\circ) = 500\sqrt{2} = 707.1 \, \textrm{mph}

.

Vector Cross Product

Assume that we have vectors a and b defined as

$\begin{align} \vec{a} &= a_x \hat{i} + a_y \hat{j} + a_z \hat{k} \\ \vec{b} &= b_x \hat{i} + b_y \hat{j} + b_z \hat{k} \end{align}$

where $\hat{i}$ , $\hat{j}$ , and $\hat{k}$ represent the unit-normal vectors in the x, y, and z directions, respectively. The cross product of a and b is then defined as

\vec{c} = \vec{a} \times \vec{b} = \left( a_y b_z -a_z b_y \right) \hat{i} + \left(a_z b_x -a_x b_z\right) \hat{j} + \left(a_x b_y -a_y b_x\right) \hat{k}

Matrix-Vector Product

Matrix-Matrix Product

Linear Systems of Equations

Solving Linear Systems of Equations

This section is a stub and needs to be expanded.

If you can provide information or finish this section you're welcome to do so and then remove this message afterwards.

@@ Line 1: / Line 1: @@
-{{Stub|page}}
+== Basics ==
+Let's first make some definitions:
-== Basics: Matrices & Vectors ==
+; Vector : A one-dimensional collection of numbers. Examples:
+: <math>a=\left( \begin{array}{cccc} 1 & 2 & 3 & 4 \end{array} \right), \quad b=\left( \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right)</math>
- Discussion of a matrix and vector - what are they and how are they formed.
+; Matrix : A two-dimensional collection of numbers. Examples:
+: <math> A = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right], \quad B = \left[ \begin{array}{cc} 9 & 6 \\ 2 & 3 \end{array}\right]</math>
-See [[Matlab_Arrays|here]] for help on creating matrices & vectors in MATLAB
+; Array : An ''n''-dimensional collection of numbers.  A ''vector'' is a 1-dimensional array while a ''matrix'' is a 2-dimensional array.
+; Transpose : An operation that involves interchanging the rows and columns of an array.  It is indicated by a <math>^\mathsf{T}</math> superscript. For example:
+: <math>a=\left( \begin{array}{cccc} 1 & 2 & 3 & 4 \end{array} \right) \; \Rightarrow \; a^\mathsf{T} = \left( \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right) </math>
+: <math> A = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right] \; \Rightarrow \;
+  A^\mathsf{T} = \left[ \begin{array}{cc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array} \right]
+</math>
+; Vector Magnitude, <math>\left|\vec{a}\right|</math> : A measure of the length of a vector, <math>\left|\vec{a}\right| = \sqrt{ \sum_{i=1}^{n} a_i^2 }</math>
+: Example: <math>\vec{a}=\left( \begin{array}{ccc} 2 & 8 & 3 \end{array} \right) \; \Rightarrow \; \left|\vec{a}\right| = \sqrt{ 2^2 + 8^2 + 3^2 }</math>
+: Example: <math>\vec{b}=\left( \begin{array}{cccc} b_1 & b_2 & b_3 & b_4\end{array} \right) \; \Rightarrow \; \left|\vec{b}\right| = \sqrt{ b_1^2 + b_2^2 + b_3^2 +b_4^2 } </math>
 == Matrix & Vector Algebra ==
+There are many common operations involving matrices and vectors including:
+* [[#Vector_Dot_Product|Vector dot product]]
+* [[#Vector_Cross_Product|Vector cross product]]
+* [[#Matrix-Vector Product|Matrix-vector product]]
+* [[#Matrix-Matrix Product|Matrix-matrix product]]
+These are each discussed in the following sub-sections.
 === Vector Dot Product ===
+The dot product of two vectors produces a scalar.  Physically, the dot product of '''a''' and '''b''' represents the ''projection'' of '''a''' onto '''b'''.
+Given two vectors '''a''' and '''b''', their dot product is formed as
+<center><math>
+ c = \vec{a} \cdot \vec{b} = \sum_{i=1}^n a_i b_i,
+</math></center>
+where <math>a_i</math> and <math>b_i</math> are the components in vectors '''a''' and '''b''' respectively.  This is most useful when we know the components of the vectors '''a''' and '''b'''.
+Occasionally we know the magnitude of the two vectors and the angle between them.  In this case, we can calculate the dot product as
+<center><math>
+ |c| = |a|\,|b|\,\cos(\theta),
+</math></center>
+where <var>&theta;</var> is the angle between the vectors '''a''' and '''b'''.
+For more information on the dot product: [http://en.wikipedia.org/wiki/Dot_product]
+==== Thought examples ====
+Consider the cartoon shown to the right.  [[image:dotProdExample.png|right|200px]]
+* At noon, when the sun is directly overhead, a rocket is launched directly toward the sun at 1000 mph.  How fast is its shadow moving?
+: Define the rocket's velocity as <math>\vec{R}</math>.
+: Define the unit normal on the ground (i.e. the direction of the rocket's shadow on the ground) as <math>\vec{s}</math>, where <math>|\vec{s}| = 1</math>.
+: Intuition tells us that if the it is moving directly toward the sun, then its shadow does not appear to move at all.  : The dot product <math>\vec{R}\cdot\vec{s} = |\vec{R}| \, |\vec{s}| \cos(\theta) = 0</math> since cos(90&deg;)=0.
+* Consider the same rocket going parallel to the earth's surface.  How fast is its shadow moving?
+: If the rocket is going parallel to the earth's surface, our intuition tells us that is shadow is moving at the same speed.  This is confirmed by the mathematics, since the angle between the rocket's path and the ground is 0. Therefore, <math>\vec{R}\cdot\vec{s} = |\vec{R}| \, |\vec{s}| \cos(\theta) = 0</math> since cos(0&deg;)=1.
+* What if the rocket were going at a 45&deg; angle?
+: Our intuition tells us that the shadow will appear to move, but it will not be moving as fast as the rocket is moving.  Mathematically, we have <math>v_s = \vec{R}\cdot\vec{s} = |\vec{R}| |\vec{s}| \cos(45^\circ) = 500\sqrt{2} = 707.1 \, \textrm{mph}</math>.
 === Vector Cross Product ===
+Assume that we have vectors '''a''' and '''b''' defined as
+<center>
+<math>
+\begin{align}
+ \vec{a} &= a_x \hat{i} + a_y \hat{j} + a_z \hat{k} \\
+ \vec{b} &= b_x \hat{i} + b_y \hat{j} + b_z \hat{k}
+\end{align}
+</math>
+</center>
+where <math>\hat{i}</math>, <math>\hat{j}</math>, and <math>\hat{k}</math> represent the unit-normal vectors in the <var>x</var>, <var>y</var>, and <var>z</var> directions, respectively.
+The cross product of '''a''' and '''b''' is then defined as
+<center><math>
+ \vec{c} = \vec{a} \times \vec{b} = \left( a_y b_z -a_z b_y \right) \hat{i} + \left(a_z b_x -a_x b_z\right) \hat{j} + \left(a_x b_y -a_y b_x\right) \hat{k}
+</math></center>
 === Matrix-Vector Product ===
 === Matrix-Matrix Product ===

Difference between revisions of "Linear Algebra"

Revision as of 09:36, 25 August 2008

Contents

Basics

Matrix & Vector Algebra

Vector Dot Product

Thought examples

Vector Cross Product

Matrix-Vector Product

Matrix-Matrix Product

Linear Systems of Equations

Solving Linear Systems of Equations

Direct Solvers

Gaussian Elimination

The Thomas Algorithm (Tridiagonal Systems)

Iterative Solvers

Jacobi Method

Gauss-Seidel Method

Other Methods

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