Basics: Vectors and Matrices

• Matlab is an interactive system for numerical calculations and visualization. It's basic element is the twodimensional array of floating point numbers, the matrix:

      A=[1 2 3; -1 0 2; 4 5 6]
     
     
     A =
     
          1     2     3
         -1     0     2
          4     5     6

• Vectors are just arrays with dimension 1xn (row vectors):

      B = [1 3 5]
     
     B =
     
          1     3     5

• or - using the matrix transposition -column vectors:

      B'
     
     ans =
     
          1
          3
          5

• We can now easily solve the system of linear equations A*X = B', using Matlab's "division" operator:

      X = A\ B'
     
     X =
     
         6.3333
        -9.6667
         4.6667

• and test the result by applying A to X:

      A*X
     
     ans =
     
         1.0000
         3.0000
         5.0000

• Larger systems are solved as easily, e.g. the following with random coefficients

      A=rand(1000);
      B=rand(1000,1);
      X = A\ B;

• takes half a minute to solve on a (not too old) PC. We check the result by looking at the largest error:

      error = max(abs(A*X-B))
     
     error =
     
       1.9213e-012