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Description

A detailed tutorial on projections. Step by step tutorial including several examples of what a projection is for reference.

Overview

A projection is another term for a transformation. But a projection is a different kind of transformation than a real transformation is. A projection is a transformation of points and lines from one plane to another plane. This is done by connecting corresponding points on the planes with parallel lines. Typically projections are used with vectors, which are entirely composed of points and lines.

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A detailed tutorial on linear transformations. Step by step tutorial including several examples of linear transformations for reference.

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

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A detailed tutorial on how to find the determinant. Step by step tutorial including several examples of finding the determinant for reference.

The determinant is a number that is associated with a square matrix. In a mathematical sense, the determinant is a scale factor for measure when the matrix is regarded as a linear transformation. The determinant is denoted by two bars on either side of the matrix, which can be confused with the absolute value of the matrix. The determinant is found by subtracting the products of the diagonals of the matrix, at least in a 2×2 matrix.

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A detailed tutorial on summation by parts. Step by step tutorial including several examples of summation by parts for reference.

Summation by parts transforms the summation of products of sequences into other summations. Often it will simplify the computation of certain sums. Summation by parts is also referred to as Abel’s lemma or Abel’s transformation. Summation by parts is similar to integration by parts, only by using summation instead of integration. In mathematical notation, summation by parts can be written as: .

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A detailed tutorial on Lorentz transformations. Step by step tutorial including several examples of Lorentz transformations for reference.

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.