Posts Tagged ‘space’
Thursday, December 17th, 2009
The Story of the Infinite Hotel
Description
A detailed tale of the Infinite Hotel. Step by step story including several pictures and an explanation of the Infinite Hotel for reference.
Overview
The Infinite Hotel is a famous math story and puzzle that was thought of by David Hilbert, a German mathematician. Sometimes the Infinite Hotel is called Hilbert’s Paradox of the Grand Hotel. It states that if one person comes into the hotel and all the rooms are full, they can all move down one room and the person can then take the first room. If k number of people come into the hotel and all the rooms are full, everyone can move down k number of rooms to make room for the people that just arrived. And, if double the amount of people that are already there are looking for rooms, everyone in room n can move to room 2n, making room for all the new arrivals in the odd-numbered rooms. This example of the Infinite Hotel can be used in certain forms of mathematical induction, and also in set theory and studies dealing with infinite numbers.
Tags: algebra, arrivals, David Hilbert, double, down, German, grand, Hilbert, hotel, induction, infinite, infinity, k, move, n!, new, numbers, paradox, room, set, space, theory
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Tuesday, November 17th, 2009
Introduction to Orthogonal Vectors
Description
A detailed tutorial on orthogonal vectors. Step by step tutorial including several examples of orthogonal vectors for reference.
Overview
Orthogonal vectors are vectors that are perpendicular. You can determine if vectors are perpendicular by finding the dot product. If the dot product is equal to zero, then the vectors are perpendicular. In certain dimensions, it is possible for three vectors to be perpendicular to each other. In this case, all three of those vectors are considered to be orthogonal. However, in general, orthogonal vectors is a term used to describe a pair of vectors.
Tags: algebra, dot, linear, pair, perpendicular, product, space, three, three-space, two, vectors, zero
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Friday, November 6th, 2009
Overview of Orthogonal Complements
Description
A detailed tutorial on orthogonal complements. Step by step tutorial including several examples of orthogonal complements for reference.
Overview
The orthogonal complement of a subspace of an inner product space is the set of all vectors in the inner product space that are orthogonal to every vector in the subspace. This can be expressed mathematically in the formula 
, where W is the subspace and V is the inner product space. The orthogonal complement is sometimes also called the perpendicular complement, shortened to the informal form perp.
Tags: algebra, complement, formula, inner, orthogonal, perp, perpendicular, product, set, space, subspace, v, vector, W
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Friday, November 6th, 2009
Introduction to Scalars
Description
A detailed tutorial on what a scalar is. Step by step tutorial including several examples of scalars and how they relate to vectors for reference.
Overview
A scalar is a number that relates vectors on a vector space through the process of scalar multiplication. A scalar can be taken from any set of numbers, including rational, algebraic, real, and complex sets of numbers. The scalar is always a real number. A scalar is a single component, and things such as vectors, matrices, and tensors can be reduced to a scalar.
Tags: algebra, algebraic, complex, component, compound, matrices, matrix, number, quaternions, rational, real, scalar, single, space, tensor, vector
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Thursday, November 5th, 2009
Linear Subspaces Explained
Description
A detailed tutorial on linear subspaces and how to identify linear subspaces. Step by step tutorial including several examples of linear subspaces for reference.
Overview
A linear subspace is usually referred to as simply a subspace, when it needs to be distinguished from other types of subspaces. Linear subspaces are also sometimes referred to as vector subspaces. In mathematical terms, to identify a linear subspace, we say that K is a field (or a set, like of real numbers), and V is a vector space over K. Elements of V are vectors and elements of K are scalars. W is said to be a subset of V. If W is a vector space itself, with the same vector space operations as V, then it has a subspace of V.
Tags: algebra, element, field, k, linear, number, operations, real, scalar, set, space, subset, subspace, v, vector, W
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Thursday, November 5th, 2009
Introduction to Linear Transformations
Description
A detailed tutorial on linear transformations. Step by step tutorial including several examples of linear transformations for reference.
Overview
A linear transformation takes place between two vector spaces. For two vector spaces V and W, there is a map T such that T(v_1 + v_2) = T(v_1) + T(v_2) for any vectors v_1 and v_2 in V, and T(a v) = a T(v) for any scalar a. Examples of linear transformation are often obtained through matrix multiplication. Linear transformations can also be injective or surjective
Tags: algebra, injective, linear, map, matrix, multiplication, scalar, space, surjective, transformation, vector
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Friday, October 30th, 2009
How to Solve Lorentz Transformations
Description
A detailed tutorial on Lorentz transformations. Step by step tutorial including several examples of Lorentz transformations for reference.
Overview
A Lorentz transformation is a way of describing how two different measurements of space and time can be converted into one frame of reference. This is because it was discovered that people who are moving at different velocities will report different times of certain events, or even a different order of events. The speed or velocity at which they are moving will throw things off. So by using a Lorentz transformation, you can get two different accounts to match up. Typically, a Lorentz transformation is a linear transformation.
Tags: algebra, events, frame, linear, Lorentz, measurement, order, reference, space, speed, time, transformation, velocity
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Tuesday, October 27th, 2009
Introduction to Minkowski Space
Description
A detailed tutorial of the application of Minkowski space. Step by step tutorial including several examples of Minkowski space for reference.
Overview
Minkowski space, sometimes referred to as Minkowski spacetime, is the setting in which Einstein’s theory of relativity was formed. Three ordinary dimensions of space are combined with a single dimension of time. This makes Minkowski space a four-dimensional manifold for representing spacetime. Minkowski space is often contrasted with Euclidean space because they are the same, except that Euclidean space has no dimension of time, and Minkowski space does.
Tags: algebra, dimension, Einstein, Euclidean, manifold, Minkowski, relativity, space, spacetime, theory, time
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Tuesday, October 27th, 2009
Introduction to Hilbert Space
Description
A detailed tutorial on on the application of Hilbert space. Step by step tutorial including several examples of Hilbert space for reference.
Overview
A Hilbert space is commonly used in vector algebra and calculus to generalize the notion of Euclidean space. It is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Hilbert spaces are also complete, which is a property that allows enough limits in the space for calculus to be used accurately.
Tags: abstract, algebra, angle, Calculus, complete, Hilbert, inner, length, limit, product, space, structure, vector
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Tuesday, October 27th, 2009
Definition of a Null Vector
Description
A detailed tutorial on the definition of a null vector. Step by step tutorial including several examples of null vectors for reference.
Overview
A null vector is a vector that has no direction. It is placed at the coordinates (0, 0, 0) in Euclidean space. Another name for a null vector is a zero vector. Although the null vector is the only vector that has no direction, we cannot say that the null vector is unique because more than one vector has the possibility of being null.
Tags: 0, algebra, arrow, coordinates, direction, Euclidean, length, magnitude, null, space, vector, zero
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