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Week 1#

Lecturer: G Venkiteswaran, Faculty for BITS Pilani
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Date: 25/Jul/2021

Topics Covered#

  1. Matrices and their types
  2. REF and RREF
  3. Rank, its computation and properties
  4. Determinent, it's computation and properties
  5. Consistency and inconsistency of linear systems
  6. Modelling using linear equations
  7. Vector and matrix norms

Matrices and their types#

A rectangular array of numbers or functions enclosed in brrackets are considered as matrices (Matrix for singular). All of the following are valid matrices:

\[A = \begin{bmatrix}a & b & c & d\\e & f & g & h\end{bmatrix},B = \begin{bmatrix}a & b & c\\d & e & f\\g & h & i\end{bmatrix},C = \begin{bmatrix}a\\b\\c\end{bmatrix},D = \begin{bmatrix}a & b & c\end{bmatrix}\]

Each element within the matrix (\(a\), \(b\), \(c\) etc) are called entities
Each matrix has a size that is represented as \(n\ x\ m\) where \(n\) is the number of rows and \(m\) is the number of columns. In the above example the array \(A\) has a size of \(2\ x\ 4\)

out of the last two matrices in the above example the matrix \(C\) is what we call a column matrix and \(D\) is called a row matrix. The other word for it is column/row Vector and Vectors are usually denoted with lower case characters like this:

\[a = \begin{bmatrix}1\\2\\3\end{bmatrix},b = \begin{bmatrix}4 & 5 & 6\end{bmatrix}\]

Two matrices are equal if both have the same size (same number of rows and columns) and every element in one matrix matches with every element in the other (including their positions)

Matrix operations#

  • Addition: Two matrices can be added if both the matrices have the same size. The addition is merely adding the same values that have the same position on both matrices. Consider the following example
\[ \begin{aligned} A = \begin{bmatrix}1 & 2 & 3 & 4\\5 & 6 & 7 & 8\end{bmatrix}, B = \begin{bmatrix}9 & 10 & 11 & 12\\13 & 14 & 15 & 16\end{bmatrix} \\ \\ C = A + B = \begin{bmatrix}1 & 2 & 3 & 4\\5 & 6 & 7 & 8\end{bmatrix} + \begin{bmatrix}9 & 10 & 11 & 12\\13 & 14 & 15 & 16\end{bmatrix} \\ \\ C = \begin{bmatrix}1 + 9 & 2 + 10 & 3 + 11 & 4 + 12\\5 + 13 & 6 + 14 & 7 + 15 & 8 + 16\end{bmatrix} \\ \\ C = \begin{bmatrix}10 & 12 & 14 & 16\\18 & 20 & 22 & 24\end{bmatrix} \end{aligned} \]

Addition adhere to the following rules:

\[ A + B = B + A \]
\[ (A + B) + C = A + (B + C) \]
\[ A + 0 = A \]
\[ A + (-A) = 0 \]
  • Multiplication: Two matrices can only be multiplied if the number of columns in the first part of the product equals the number of rows in the second part of the product to yield a product matrix with a size of the number of rows of the first part and the number of columns of the second part. The size of matrices after a product would look like this:
\[A_(m*p) * B_(p*n) = C_(m*n)\]

The formulae for calculating the entities within the product matrix are as follows, given that \(i\) and \(j\) are the row and column number that identifies element of that matrix:

\[A\ is\ an\ array\ with\ a_{ij}\ for\ entities\ and\ have\ size\ m*n\]
\[B\ is\ an\ array\ with\ b_{ij}\ for\ entites\ and\ have\ size\ n*p\]
\[C\ is\ the\ product\ array\ with\ c_{ij}\ for\ entites\ and\ have\ size\ n*m, then\]
\[c_{ij} = \begin{equation}\sum_{k = 1}^{n}a_{ik}b_{kj}\end{equation}\]
\[where\ j = 1, . . . , m\ and\ k = 1, . . . , p\]

Tags: !MathematicalFoundationsIndex