Applied Mathematics/The Basics

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The Basics of linear algebra[edit]

 \mathbf{A} = 
 \begin{bmatrix}
 a_{11} & a_{12} & a_{12} & a_{14} \\
 a_{21} & a_{22} & a_{23} & a_{24} \\
 a_{31} & a_{32} & a_{33} & a_{34} \\
  \end{bmatrix}.

A matrix is composed of a rectangular array of numbers arranged in rows and columns. The horizontal lines are called rows and the vertical lines are called columns. The individual items in a matrix are called elements. The element in the i-th row and the j-th column of a matrix is referred to as the i,j, (i,j), or (i,j)th element of the matrix. To specify the size of a matrix, a matrix with m rows and n columns is called an m-by-n matrix, and m and n are called its dimensions.

Basic operation[1][edit]

Operation Definition Example
Addition The sum A+B of two m-by-n matrices A and B is calculated entrywise:
(A + B)i,j = Ai,j + Bi,j, where 1 ≤ im and 1 ≤ jn.



\begin{bmatrix}
1 & 3 & 1 \\
1 & 0 & 0
\end{bmatrix}
+
\begin{bmatrix}
0 & 0 & 5  \\
7 & 5 & 0
\end{bmatrix}
=
\begin{bmatrix}
1+0 & 3+0 & 1+5 \\
1+7 & 0+5 & 0+0
\end{bmatrix}
=
\begin{bmatrix}
1 & 3 & 6 \\
8 & 5 & 0
\end{bmatrix}

Scalar multiplication The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c:
(cA)i,j = c · Ai,j.
2 \cdot

\begin{bmatrix}
1 & 8 & -3 \\
4 & -2 & 5
\end{bmatrix}
=
\begin{bmatrix}
2 \cdot 1 & 2\cdot 8 & 2\cdot -3 \\
2\cdot 4 & 2\cdot -2 & 2\cdot 5
\end{bmatrix}
=
\begin{bmatrix}
2 & 16 & -6 \\
8 & -4 & 10
\end{bmatrix}
Transpose The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:
(AT)i,j = Aj,i.


\begin{bmatrix}
1 & 2 & 3 \\
0 & -6 & 7
\end{bmatrix}^\mathrm{T} =
\begin{bmatrix}
1 & 0 \\
2 & -6 \\
3 & 7
\end{bmatrix}

Practice problems[edit]

(1) 
\begin{bmatrix}
 5 & 7 & 3 \\
 1 & 2 & 9
\end{bmatrix}	
+
\begin{bmatrix}
4 & 0 & 5  \\
8 & 3 & 0
\end{bmatrix}	
=
(2) 4 
\begin{bmatrix}
-1 & 0 & -5 \\
7 & 9 & -6 	
\end{bmatrix}
=
(3) 
\begin{bmatrix}
-2 & 5 & 7 \\
0 & 0 & 9
\end{bmatrix}^\mathrm{T}	
=

Matrix multiplication[edit]

Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B[2]


 [\mathbf{AB}]_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + \cdots + A_{i,n}B_{n,j} = \sum_{r=1}^n A_{i,r}B_{r,j}[3]


Schematic depiction of the matrix product AB of two matrices A and B.



Example[edit]

 
 
	
\begin{bmatrix}
 
	
-2 & 0 \\	
3 & 2
 	
\end{bmatrix}
 		 
	
\begin{bmatrix}
 
	
1 & 2 \\
 	
3 & -1
 	
\end{bmatrix}



	
= \begin{bmatrix}
-2+0 & -4+0 \\
3+6 & 6+(-2)
\end{bmatrix}

	
=\begin{bmatrix}
-2 & -4 \\
9 & 4
\end{bmatrix}

Practice Problems[edit]

(1)  	
\begin{bmatrix}
1 & 0 \\
2 & 2
\end{bmatrix}
 
\begin{bmatrix}	
4 \\
2	
\end{bmatrix}	
=

(2)  	
\begin{bmatrix}
1 & 2 \\
2 & 3
\end{bmatrix}

\begin{bmatrix}	
2 & 3 \\
1 & 4
\end{bmatrix}
=


Dot product[edit]

A row vector is a 1 × m matrix, while a column vector is a m × 1 matrix.

Suppose A is row vector and B is column vector, then the dot product is defined as follows;

	A \cdot B= |A||B| cos \theta

or

\mathbf{A}\cdot \mathbf{B} =
\begin{pmatrix}a_1 & a_2 & \cdots & a_n\end{pmatrix}
\begin{pmatrix}b_1 \\ b_2 \\ \vdots \\ b_n\end{pmatrix}
= a_1b_1+a_2b_2+\cdots+a_nb_n = \sum_{i=1}^n a_ib_i


Suppose 
\mathbf{A} = 
\begin{pmatrix}a_1 & a_2 & a_3\end{pmatrix}
and 	
\mathbf{B} = 
\begin{pmatrix}b_1 \\ b_2 \\ b_3\end{pmatrix}
The dot product is

\mathbf{A}\cdot \mathbf{B} =	
\begin{pmatrix}a_1 & a_2 & a_3\end{pmatrix}
\begin{pmatrix}b_1 \\ b_2 \\ b_3\end{pmatrix}
= a_1b_1+a_2b_2+a_3b_3

Example[edit]

Suppose 
\mathbf{A} = 
\begin{pmatrix}2 \\ 1 \\ 3\end{pmatrix}
and 
\mathbf{B} = 
\begin{pmatrix}7 \\ 5 \\ 4\end{pmatrix}

\mathbf{A}\cdot \mathbf{B} =
\begin{pmatrix}2 & 1 & 3\end{pmatrix}
\begin{pmatrix}7 \\ 5 \\ 4\end{pmatrix}

= 2\cdot7+1\cdot5+3\cdot4
=14+5+12

=31

Practice problems[edit]

(1) \mathbf{A} = 
\begin{pmatrix}3 \\ 2 \\ 5\end{pmatrix}
and 
\mathbf{B} = 
\begin{pmatrix}1 \\ 4 \\ 3\end{pmatrix}

\mathbf{A}\cdot \mathbf{B} =

(2) \mathbf{A} = 
\begin{pmatrix}1 \\ 0 \\ 3\end{pmatrix}
and 
\mathbf{B} = 
\begin{pmatrix}6 \\ 9 \\ 2\end{pmatrix}

\mathbf{A}\cdot \mathbf{B} =

Cross product[edit]

Cross product is defined as follows:

A \times B = |A||B| sin \theta

Or, using detriment,

\mathbf{A \times B}=\begin{vmatrix}
e_x&e_y&e_z\\
a_x&a_y&a_z\\
b_x&b_y&b_z\\
\end{vmatrix}
=
(a_y b_z - a_z b_y, a_z b_x - a_x b_z, a_x b_y - a_yb_x)

where e is unit vector.

Reference[edit]

  1. Sourced from Matrix (mathematics), Wikipedia, 28th March 2013.
  2. Sourced from Matrix (mathematics), Wikipedia, 30th March 2013.
  3. Sourced from Matrix (mathematics), Wikipedia, 30th March 2013.