Linear Systems and Matrices

Linear Systems

An n by n linear system of equations is a system of n linear equations in n variables.

        a₁₁x₁ + a₁₂x₂ + ... + a_1nx_n  =  b₁
        a₂₁x₁ + a₂₂x₂ + ... + a_2nx_n  =  b₂ 
        ...            ...                ...        ...
        a_n1x₁ + a_n2x₂ + ... + a_nnx_n  =  b_n 

Example

Solve

        2x₁ + 3x₂  =  9
         x₁ - 2x₂  =  1

Solution

To solve this we sequentially perform members of the following three operations:

1. Switch two equations.
   
   
2. Multiply an equation by a nonzero constant.
   
   
3. Replace an equation by that equation plus a multiple of the second equation.
   

We have

                                    Switching the two equations
         x₁ - 2x₂  =  1            
        2x₁ + 3x₂  =  9

                                    Replace the 2nd equation with the 2nd equation + (-2)1st equation
         x₁ - 2x₂  =  1          
                7x₂  =  7

                                    Multiply the second equation by 1/7
         x₁ - 2x₂  =  1          
                  x₂  =  1

                                    Replace the 1st equation with the 1st equation + (2)2nd equation

         x₁   =  3          
         x₂  =  1

Matrices

An m by n matrix is an array of numbers with m rows and n columns. 

Example

The matrix below is a 2 by 3 matrix.

A square matrix is an n by n matrix, that is a matrix such that the number of rows is equal to the number of columns.  The ij^th entry is the number in the i^th row and j^th column.  For example, the the matrix above the 1 2^th entry is

        a₁₂  =  4

Note:  A vector such at <2,4,6> can be looked as a 1 by 3 matrix.  

A square matrix is called a diagonal matrix if 

        a_ij  =  0        for        i  j

The matrix below is a diagonal matrix

If all the entries of a diagonal matrix are equal, then the matrix is called a scalar matrix.  The example below is a scalar matrix.

Addition Subtraction and Scalar Multiplication

Just as with vectors we can add and subtract matrices and multiply a matrix by a scalar.  To add or subtract matrices the dimensions of the two matrices must be the same.

Example

Let 

then

Two matrices are called equal if all of their entries are equal.

If A is an m by n matrix, then the transpose of A, A^T, is the n by m matrix with the rows and columns switched.

        (A^T)_ij  =  A_ji 

In the above example

Example of the Theoretical Exercise

Prove that  

        (A^T)^T  =  A

Solution

We have 

        ((A^T)^T )_ij =  (A^T)ji  =  A_ij 

Since the ij^th entries are equal for each ij, the matrices are equal.

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