Matrix algebra
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Matrices
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A matrix is an array of numbers. For example, [Equation goes here - download the original to see it.] Is a matrix. Matrices are useful because they store useful information. They also represent transformations of space. Matrices can be added, subtracted and multiplied. Addition and subtraction of matrices. We add subtract matrices by adding or subtracting their corresponding elements. It follows that it is only possible to add and subtract matrices that have the same form - the same number of rows and columns. For example: [Equation goes here - download the original to see it.] But it is not possible to add [Equation goes here - download the original to see it.] Because they have different forms Multiplication of matrices We learn first the multiplication of a square 2 x 2 matrix by a 2 x 1 column matrix that represents a vector. [Equation goes here - download the original to see it.] Let us try to visualise this process: - The row (a,b) is going to be multiplied by the column [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Imagine lifting out the row (a,b) and slotting it besides the column [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Multiply the corresponding entries, add them together and place the result in the first entry of the resultant column vector. [Diagram goes here - download the original to see it.] The overall result is [Diagram goes here - download the original to see it.] The result of the first row goes in the first entry of the column vector. Repeat the process for the second row. [Diagram goes here - download the original to see it.] The result of the second row goes in the second entry of the column vector. For Example: [Equation goes here - download the original to see it.] The multiplication of vectors can be generalised to matrices of other sizes: - [Equation goes here - download the original to see it.] We can only multiply matrices with vectors of the right form. The vector must have as many rows as the matrix has columns. For example we cannot multiply a 2x3 matrix by a 2x1 vector, because the vector is too small and we run out of terms: - [Equation goes here - download the original to see it.] The multiplication of a 2x2 matrix A by a 2x2 B Matrix is treated as the multiplication of the first matrix A by the separate columns of the matrix B. [Equation goes here - download the original to see it.] Work out [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] The second column [Diagram goes here - download the original to see it.] Thus: [Diagram goes here - download the original to see it.] For example [Equation goes here - download the original to see it.] Matrix multiplication can be generalised to other forms. For example: [Equation goes here - download the original to see it.]
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Contents of
Matrix algebra
1 Matrices
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