My Math Land: January 2014

Friday, January 31, 2014

Inconsistent systems of equations

Hey today we are going to look at how to recognize system of equations with no or infinitely many solution after we have done some consistent systems of equations!
There are four circumstances:

Two variable system of equations with Infinitely many solutions

The equations in a two variable system of equations are linear and hence can be thought of as equations of two lines. When these two lines are parallel, then the system has infinitely many solutions.

When two lines are parallel, their equations can usually be expressed as multiples of each other and that's usually a quick way to spot systems with infinitely many solutions.

For example, let's try to solve the system of equations below:

Using substitution method, we can solve for the variables as follows:

From equation (1)

substituting the above into equation(2)

In the above equation, we can see that we've lost all the variables from the equation. This means that we can pick any value of x or y then substitute it into any one of the two equations and then solve for the other variable.

For example if we pick x = 0, then if we substitute this into equation (1) we would get y = 1. Any value we pick for x would give a different value for y and thus there are infinitely many solutions for the system of equations.

Two variable systems of equations with NO SOLUTION

There also exist two variable system of equations with no solution at all. This happens when as we attempt to solve the system we end up an equation that makes no sense mathematically.

For example, solve the system of equations below:

Using matrix method we can solve the above as follows:

Reducing the above to Row Echelon form can be done as follows:

Adding row 2 to row 1:

The equation formed from the second row of the matrix is given as

which means that:

But we know that the above is mathematically impossible. When we come across the above, we say that the system of equations has NO SOLUTION. Thus we refer to such systems as being inconsistent because they don't make any mathematical sense.

Three variable systems of equations with Infinite Solutions

When discussing the different methods of solving systems of equations, we only looked at examples of systems with one unique solution set. These are known as Consistent systems of equations but they are not the only ones. Three variable systems of equations with infinitely many solution sets are also called consistent.

Since the equations in a three variable system of equations are linear, they can also be thought of as equations of planes. The way these planes interact with each other defines what kind of solution set they have and whether or not they have a solution set. When these planes are parallel to each other, then the system of equations that they form has infinitely many solutions.

Just as with two variable systems, three variable sytems have an infinte set of solutions if when you solving for the variables you end up with an equation where all the variables disappear.

For example; solve the system of equations below:

Solution:

Using matrix method:

In the last row of the above augmented matrix, we have ended up with all zeros on both sides of the equations. This means that two of the planes formed by the equations in the system of equations are parallel, and thus the system of equations is said to have an infinite set of solutions. We solve for any of the set by assigning one variable in the remaining two equations and then solving for the other two.

Three variable systems with NO SOLUTION

Three variable systems of equations with no solution arise when the planed formed by the equations in the system neither meet at point nor are they parallel. As a result, when solving these systems, we end up with equations that make no mathematical sense.

Thursday, January 30, 2014

8.1 matrix

IHey guys today lets do some system of linear equations using Gaussian elimination!

Steps:

1.Write the augmented matrix of the linear equations

2.use elementary row operations to rewrite the matrix in row-echelon form

3. Write the linear equations corresponding to the matrix in row-echelon form,

4. Use back-substitution to find the other variables

Now, are you ready for some problems?

Here are the two examples:

I pulled these examples from the practice problems, so you can check it out in miss v's math land!

Prezi for chapter 8 vocabularies

This is our team's prezi for chapter 8 vocabularies!

http://www.prezi.com/cv8ct5-kvqol/

Citation for the pictures:

http://0.tqn.com/d/visualbasic/1/0/w/L/GDIP05-01.gif

http://www.mathwarehouse.com/algebra/matrix/images/matrix-image.gif

http://upload.wikimedia.org/wikipedia/commons/b/bb/Matrix.svg image for columns and rows

http://upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Matrix_Columns.svg/510px-Matrix_Columns.svg.png

http://www.icoachmath.com/image_md/Main%20Diagonal1.jpg

http://www.math.tamu.edu/~stecher/Linear-Algebra/Systems/eq-matrix2.gif

http://algebra.freehomeworkmathhelp.com/Systems/Matrix_Algebra/coefficient_matrix.PNG

http://www.mathsisfun.com/algebra/images/matrix-gauss-jordan2.gif

http://www.math.nyu.edu/~neylon/linalgfall04/project1/jja/invalgo.jpg

http://i.stack.imgur.com/ST1qx.png

https://www.google.com/search?client=safari&hl=en&biw=1024&bih=672&tbm=isch&sa=1&ei=CkDpUquHIMHcoATF84KYDA&q=leading+1+in+matrix&oq=leading+1+in+&gs_l=img.1.0.0i24.2384.3432.0.5714.4.3.0.1.1.0.411.624.0j2j4-1.3.0....0...1c.1.32.img..0.4.631.nJ3MPaP2Tcw#facrc=_&imgrc=UEnP7PCILcdD6M%253A%3Bi7yYKAstZcZ8fM%3Bhttp%253A%252F%252Fupload.wikimedia.org%252Fmath%252Fe%252F7%252F0%252Fe702c4415f74a443a8827e9d120a30bd.png%3Bhttp%253A%252F%252Fen.wikipedia.org%252Fwiki%252FList_of_matrices%3B180%3B109

http://www.efunda.com/math/num_linearalgebra/images/GaussianElimination02.gif

https://www.google.com/search?q=infinite+solutions&client=safari&hl=en&source=lnms&tbm=isch&sa=X&ei=eEDpUpuAIs_poASzs4DoCA&ved=0CAkQ_AUoAA&biw=1024&bih=672#facrc=_&imgrc=_lglgOzcQE52SM%253A%3Bp98D7MGVF3pz3M%3Bhttp%253A%252F%252Fwww.jamesbrennan.org%252Falgebra%252Fsystems%252Fsolution_set_files%252Fimage006.gif%3Bhttp%253A%252F%252Fwww.jamesbrennan.org%252Falgebra%252Fsystems%252Fsolution_set.htm%3B274%3B235

http://oregonstate.edu/instruct/ch590/lessons/lesson13_files/image002.jpg

hodor.org/unchem-old/math/matrix/determinant.gif

http://www.mathwarehouse.com/algebra/matrix/images/square-matrix/inverse-matrix.gif

http://mathworld.wolfram.com/images/equations/Minor/NumberedEquation1.gif

http://www.mathwords.com/c/c_assets/c46a.gif

http://upload.wikimedia.org/wikipedia/commons/thumb/6/63/Cyclic_group_Z4%3B_Cayley_table%3B_powers_of_Gray_code_permutation_(small).svg/478px-Cyclic_group_Z4%3B_Cayley_table%3B_powers_of_Gray_code_permutation_(small).svg.png

Cramer's Rule: an explicit formula for the solution of a system of linear quations with as many equations as unknowns, valid whenever the system has a unique solution.

http://www.statisticslectures.com/images/

http://www.icoachmath.com//image_md/Collinear%20Points1.jpg

Cryptogram: a type of puzzle that consists of a short piece of encrypted text

http://upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Cryptogram-example.jpg/297px-Cryptogram-example.jpg

http://upload.wikimedia.org/wikipedia/commons/f/f8/Crypto.png

Monday, January 27, 2014

Mathematical symbols- do u know these abbreviations?

Today lets learn about math symbols!!

Some of them may be really easy, but just bear with them!

Symbol in HTML	Symbol in $T E X$	Name	Explanation	Examples
		Read as
		Category
=	$=$	equality is equal to; equals everywhere	$x = y$ means $x$ and $y$ represent the same thing or value.	$2 = 2$ $1 + 1 = 2$
≠	$\ne$	inequality is not equal to; does not equal everywhere	$x \ne y$ means that $x$ and $y$ do not represent the same thing or value. (The forms !=, /= or <> are generally used in programming languages where ease of typing and use of ASCII text is preferred.)	$2 + 2 \ne 5$
< >	$<$ $>$	strict inequality is less than, is greater than order theory	$x < y$ means $x$ is less than $y$ . $x > y$ means $x$ is greater than $y$ .	$3 < 4$ $5 > 4$
< >	$<$ $>$	proper subgroup is a proper subgroup of group theory	$H < G$ means $H$ is a proper subgroup of $G$ .	$5Z < Z$ $A_3 < S_3$
≪ ≫	$\ll \!\,$ $\gg \!\,$	(very) strict inequality is much less than, is much greater than order theory	x ≪ y means x is much less than y. x ≫ y means x is much greater than y.	0.003 ≪ 1000000
≪ ≫	$\ll \!\,$ $\gg \!\,$	asymptotic comparison is of smaller order than, is of greater order than analytic number theory	f ≪ g means the growth of f is asymptotically bounded by g. (This is I. M. Vinogradov's notation. Another notation is the Big O notation, which looks like f = O(g).)	x ≪ e^x
≤ ≥	$\le \!\,$ $\ge$	inequality is less than or equal to, is greater than or equal to order theory	x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to y. (The forms <= and >= are generally used in programming languages, where ease of typing and use of ASCII text is preferred.)	3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5
		subgroup is a subgroup of group theory	H ≤ G means H is a subgroup of G.	Z ≤ Z A₃ ≤ S₃
		reduction is reducible to computational complexity theory	A ≤ B means the problem A can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction.	If $\exists f \in F \mbox{ . } \forall x \in \mathbb{N} \mbox{ . } x \in A \Leftrightarrow f(x) \in B$ then $A \leq_{F} B$
≦ ≧	$\leqq \!\,$ $\geqq \!\,$	congruence relation ...is less than ... is greater than... modular arithmetic	7k ≡ 28 (mod 2) is only true if k is an even integer. Assume that the problem requires k to be non-negative; the domain is defined as 0 ≦ k ≦ ∞.	10a ≡ 5 (mod 5) for 1 ≦ a ≦ 10
≦ ≧	$\leqq \!\,$ $\geqq \!\,$	vector inequality ... is less than or equal... is greater than or equal... order theory	x ≦ y means that each component of vector x is less than or equal to each corresponding component of vector y. x ≧ y means that each component of vector x is greater than or equal to each corresponding component of vector y. It is important to note that x ≦ y remains true if every element is equal. However, if the operator is changed, x ≤ y is true if and only if x ≠ y is also true.
≺	$\prec \!\,$	Karp reduction is Karp reducible to; is polynomial-time many-one reducible to computational complexity theory	L₁ ≺ L₂ means that the problem L₁ is Karp reducible to L₂.^[1]	If L₁ ≺ L₂ and L₂ ∈ P, then L₁ ∈ P.
∝	$\propto \!\,$	proportionality is proportional to; varies as everywhere	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x.
∝	$\propto \!\,$	Karp reduction^[2] is Karp reducible to; is polynomial-time many-one reducible to computational complexity theory	A ∝ B means the problem A can be polynomially reduced to the problem B.	If L₁ ∝ L₂ and L₂ ∈ P, then L₁ ∈ P.