Matrix multiplication - Revision history

Alexandk at 03:55, 28 November 2021

2021-11-28T03:55:27Z

2021-11-28T03:52:33Z

2021-11-28T03:50:23Z

2021-11-28T03:43:54Z

2021-11-28T03:41:31Z

2021-11-28T03:39:28Z

2021-11-28T03:29:04Z

2021-11-28T02:40:01Z

2021-11-28T02:39:32Z

2021-11-28T02:38:00Z

← Older revision		Revision as of 03:55, 28 November 2021
Line 105:		Line 105:
	\end{cases},		\end{cases},
	</math>		</math>
		+
		+	The matrix of the last system is
		+
		+	<math>
		+	\begin{bmatrix}
		+	a_{11}b_{11} +a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\
		+	a_{21}b_{11} +a_{22} b_{21} & a_{21}b_{12}+a_{22}b_{22}
		+	\end{bmatrix},
		+	</math>
		+
		+	which is the product of matrices <math> A </math> and <math> B </math>.

@@ Line 96: / Line 96: @@
 \end{cases},
 </math>
 or, in terms of <math> y_1\text{ and } y_2 </math>:
 <math>
 \begin{cases}
-(a_{11}b_{11} +a_{12}b_{21} ) y_1 + (a_{11}b_{12}+a_{12}b_{22} )y_2+ a_{12}(b_{21} y_1 + b_{22}y_2)= c_1 \\
+(a_{11}b_{11} +a_{12}b_{21} ) y_1 + (a_{11}b_{12}+a_{12}b_{22} )y_2 = c_1 \\
-a_{21} (b_{11} y_1 + b_{12}y_2) + a_{22}(b_{21} y_1 + b_{22}y_2)= c_2
+(a_{21}b_{11} +a_{22} b_{21})y_1 + (a_{21}b_{12}+a_{22}b_{22})y_2= c_2
 \end{cases},
 </math>

@@ Line 74: / Line 74: @@
 \end{bmatrix}</math>
-Make substitution as follows:
+Let <math> B = \begin{bmatrix}
 <math>
 \begin{cases}
-x_1 = a_{11} x_1 + a_{12}x_2\\
+x_1 = b_{11} y_1 + b_{12}y_2\\
-x_2 = a_{21} x_1 + a_{22}x_2
+x_2 = b_{21} y_1 + b_{22}y_2
 \end{cases}
 </math>

@@ Line 60: / Line 60: @@
 One of the fundamental ideas in mathematics is that of change of variables (substitution), which in particular is extensively used in solving equations and systems of equations. In particular, we can use linear substitution in a system of linear equations as follows.
 \begin{cases}
 a_{11} x_1 + a_{12}x_2= c_1 \\
 a_{21} x_1 + a_{22}x_2= c_2
 \end{cases}

@@ Line 62: / Line 62: @@
 \begin{cases}
 a_{11} x_1 + a_{12}x_2= c_1 \\
-x - 2y + 4z &= 2
+a_{21} x_1 + a_{22}x_2= c_2
 \end{cases}

← Older revision		Revision as of 03:39, 28 November 2021
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	One of the fundamental ideas in mathematics is that of change of variables (substitution), which in particular is extensively used in solving equations and systems of equations. In particular, we can use linear substitution in a system of linear equations as follows.		One of the fundamental ideas in mathematics is that of change of variables (substitution), which in particular is extensively used in solving equations and systems of equations. In particular, we can use linear substitution in a system of linear equations as follows.
		+
		+	\begin{cases}
		+	a_{11} x_1 + a_{12}x_2= c_1 \\
		+	7x - 2y + 4z &= 2
		+	\end{cases}

← Older revision		Revision as of 03:29, 28 November 2021
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	'''System of linear equations'''		'''System of linear equations'''
		+
		+	One of the fundamental ideas in mathematics is that of change of variables (substitution), which in particular is extensively used in solving equations and systems of equations. In particular, we can use linear substitution in a system of linear equations as follows.

@@ Line 2: / Line 2: @@
 We start with a simple example showing a practical use of multiplying a matrix by numbers (scalars). Below is an example of a distance matrix between three cities, in kilometers:
 [[File:Distance matrix.png|alt=Distance matrix|thumb|Distance matrix]]
@@ Line 11: / Line 12: @@
 We can multiply this matrix by <math> 0.62 </math> to (approximately) convert it to distances in miles:
 <math> B =0.62\cdot A \begin{bmatrix}

@@ Line 50: / Line 50: @@
 {\color{red} 2}\cdot {\color{blue} 2}  + {\color{red} 3}\cdot {\color{blue} 5} & {\color{red} 2}\cdot {\color{blue} 3} + {\color{red} 3}\cdot {\color{blue} 1}
 \end{bmatrix} </math>
 '''System of linear equations'''