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

Vector transformations are not as difficult as one mught think – they are done just like ordinary transformations, except in terms of vectors. Rotation is one of the main types of vector transformations, and is the most common one that is done. In order for a vector to be properly transformed, they must satisfy the orthogonality condition.

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A detailed tutorial on how to define the angles between vectors. Step by step tutorial including several examples of angles between vectors for reference.

In general, it is easier to find the angle between 2D vectors, rather than 3D vectors. In order to define the angles between vectors, we need to use the dot product in conjunction with a few other functions. The angles between vectors can be expressed as angle = arccos(v1xv2), where v1xv2 is how the dot product is expressed.

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A detailed tutorial on orthogonal complements. Step by step tutorial including several examples of orthogonal complements for reference.

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.

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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.

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.

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A detailed tutorial on how to use parametrization. Step by step tutorial including several examples of how to use parametrization for reference.

Parametrization can be used in many different branches of math, including algebra and calculus. Parametrization involves setting up parameters necessary for the complete or relevent specification of a geometric object. This means it is only used when calculating a shape or part of a shape, because that is what a geometric object is. Sometimes, this is nothing more than identifying the parameters. Other times it becomes an involved mathematical process that is used to find out what the parameters are.

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A detailed tutorial on linear subspaces and how to identify linear subspaces. Step by step tutorial including several examples of linear subspaces for reference.

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.

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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 the transpose of a vector. Step by step tutorial including several examples of the transpose of a vector for reference.

The transpose of a vector is very similar to the transpose of a matrix, because even though the function the operation is being performed on changes, the operation itself doesn’t change. When you transpose a vector, it is just a way of saying the the column of your vector becomes a row, or the row of your vector becomes a column. Transposing vectors is not done very often, but it is still an important part of linear algebra.

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A detailed tutorial on the definition of a terminal point. Step by step tutorial including several examples of terminal points for reference.

A terminal point is just a way of saying the ending point. The terminal point of a line or a figure is the point where it ends. The term terminal point is used often when talking about vectors – they end at the terminal point. The terminal point is referred as the head of the vector.