Description
A detailed tutorial on finding the length of a vector. Step by step tutorial including several examples of how to find the length of a vector for reference.
Overview
The length of a vector is also known as the magnitude of a vector. This can be compared to the absolute value of a real number. In order to find the length of a vector, you need to use the Euclidean norm:
The Euclidean norm is a consequence of the Pythagorean theorem.
Description
A detailed tutorial on how to solve vectors using scalar multiplication. Step by step tutorial including several examples on scalar multiplication for reference.
Overview
Scalar multiplication is when you multiply, or re-scale, vectors by a real number. These real numbers are referred to as scalars, so that they can be distinguished from vectors. So, scalar multiplication is when you multiply a vector by a scalar. When you multiply a scalar and a vector, you will get another vector. Your resulting vector will be:
When a vector is multiplied by a scalar, the vector is getting stretched out by a factor of the scalar. If the scalar is negative, then the vector changes direction. A property of scalar multiplication is that it is distributive.
Description
A detailed tutorial on how to solve vectors using vector subtraction. Step by step tutorial including several examples of vector subtraction for reference.
Overview
Vector subtraction involves two vectors that do not have to be equal, and could have different magnitudes and directions. The vectors are referred to as a and b. The formula for vector subtraction is:
In general, vector subtraction is defined geomtrically instead of algebraically, so it is not used quite as often as vector addition is.
Description
A detailed tutorial on how to solve vectors using vector addition. Step by step tutorial including several examples of vector addition for reference.
Overview
Vector addition involves two vectors that do not have to be equal, and could have different magnitudes and directions. The vectors are referred to as a and b. The formula for vector addition is:
Vector addition is also occassionally referred to as the parallelogram rule, because on a picture diagram of vector addition the shape of a parallelogram is formed.
Description
A detailed tutorial on vector space. Step by step tutorial including several examples of vector space and how to solve for vector space for reference.
Overview
Vector space is simply a structure in mathematics that is formed by a collection of vectors. Vector space can be calculated using vector addition and scalar multiplication. Vector space is very dependent on the definition of a vector. Some vectors are simply arrows on a fixed plane. But in general, the term vector just means there is an object for which two operations can be performed. The definition of vector space is defined in algebraic terms, as opposed to the geometric terms that can sometimes be applied.
Description
A detailed tutorial on the Gram-Schmidt process. Step by step tutorial including a visual example of the Gram-Schmidt process for reference.
Overview
The Gram-Schmidt process is a process used for orthogonalizing a set of vectors in an inner product space. What the Gram-Schmidt process does is it takes a finite and linearly independent set and converts it to an orthogonal set that spans the same amount of space.
Description
A detailed tutorial on how to solve systems of equations using Cramer’s rule. Step by step tutorial including several examples of how to solve for systems of equations using Cramer’s rule for reference.
Overview
Cramer’s rule is a theorem in linear algebra that is used as an alternative method of solving systems of equations. Cramer’s rule uses matrices to solve for systems of equations, and is typically used when there is a unique solution. The solution is expressed in the form of matrices which are obtained by replacing one column of the vector of right hand sides of the equations.
