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.
Description
A detailed tutorial on the solving of the Cauchy-Schwarz Inequality. Step by step tutorial including several examples of how to solve the Cauchy-Schwarz Inequality for reference.
Overview
The Cauchy-Schwarz Inequality is also known as the Schwarz Inequality, the Bunyakovsky Inequality, and the Cauchy-Bunyakovsky-Schwarz Inequality. It was published by Augustin Cauchy and was first stated by Viktor Yakovlevich Bunyakovsky. It was later rediscovered by Hermann Amandus Schwarz. This is used mostly in linear algebra, when solving vectors. It is also used in probability theory.
The Cauchy-Schwarz Inequality states that for all vectors x and y of an inner product space,
By taking the square root of both sides, it can be written as
If the two sides are equal, and if x and y are both independent, then the formula may be restated as
Description
This video explains the difference between addition/subtraction and multiplication of matrices. It also explains why there is such a difference. Several example problems are provided in the video, along with a clear explanation of the multiplication process.
Overview
Matrix multiplication is very different from addition and subtraction with matrices. Instead of combining the numbers from the same places, you must combine rows from the first matrix with columns from the second matrix. When you are looking for the number for the top left corner – the first number of your matrix – you will look at the top row of the first matrix and the first column of the second matrix. Say you have 2 matrices like this:
{a b} * {w x}
{c d} * {y z}
You will look at a and b, and w and y, to find your first term. The first number of your new matrix will be [(a * w) + (b * y)]. When you want to find the second term – the term still in the first row, but in a different column – you will switch you focus to the other column, while keeping your row of focus the same, and solve it in the same way. This must be done for every term in the matrices.
Description
This video explains the different types and sizes of matrices and what makes up each of them. It explains how to match up the sizes so you know if addition is possible. Several sample problems for addition are provided in the video.
Overview
While with normal numbers, you can add anything, some sets of matrices cannot be added. Matrices can only be added when they are the same size. Sizes of matrices are recorded by the numbers that are in each row and colum. For instance, a matrix with 3 rows and 3 columns would be referred to as a 3×3 matrix. If your matrices are the same size, you can add them. You then simply match up the numbers in your matrices to add them. This means that the number in the top right corner of the first matrix is added to the number in the top right corner of the second matrix, and the number they add up to will go in the same place in the matrix that is the solution to the problem. Subtraction works the exact same way as addition.
Description
This video shows how to solve equations, but the most important part of this video is displayed right in the beginning – it converts a regular set of equations into a matrix, which shows you exactly what a matrix is. A sample problem is provided, or you can simply benefit from the conversion process.
Overview
A matrix, the plural of which is matrices, is a set of numbers that form an equation. They come up in many different parts of math, and are used very commonly in Linear Algebra. They make viewing a set of equations much easier. The most important thing to remember about matrices is that everything must line up. Here is an example:
1x + 2y = 18
5x + 7z = 32
Must be rewritten as:
1x + 2y + 0z = 18
5x + 0y + 7z = 32
This way it is lined up correctly when you put it in the matrix. Some matrices have a line before the solution, to seperate it from the other numbers – this is optional. Even if it is not there, the last number is always the solution. So when you put it into a matrix, it will look like this:
{1 2 0 | 18}
{5 0 7 | 32}
