.lst-kix_rdjypv61h1og-4 > li:before{content:"" counter(lst-ctn-kix_rdjypv61h1og-4,lower-latin) ". "}.lst-kix_rdjypv61h1og-5 > li:before{content:"" counter(lst-ctn-kix_rdjypv61h1og-5,lower-roman) ". "}.lst-kix_rdjypv61h1og-2 > li:before{content:"" counter(lst-ctn-kix_rdjypv61h1og-2,lower-roman) ". "}.lst-kix_rdjypv61h1og-3 > li:before{content:"" counter(lst-ctn-kix_rdjypv61h1og-3,decimal) ". "}.lst-kix_rdjypv61h1og-6 > li:before{content:"" counter(lst-ctn-kix_rdjypv61h1og-6,decimal) ". "}.lst-kix_rdjypv61h1og-7 > li:before{content:"" counter(lst-ctn-kix_rdjypv61h1og-7,lower-latin) ". "}.lst-kix_rdjypv61h1og-0 > li{counter-increment:lst-ctn-kix_rdjypv61h1og-0}ol.lst-kix_rdjypv61h1og-4.start{counter-reset:lst-ctn-kix_rdjypv61h1og-4 0}.lst-kix_rdjypv61h1og-1 > li{counter-increment:lst-ctn-kix_rdjypv61h1og-1}ol.lst-kix_rdjypv61h1og-7{list-style-type:none}.lst-kix_rdjypv61h1og-4 > li{counter-increment:lst-ctn-kix_rdjypv61h1og-4}ol.lst-kix_rdjypv61h1og-6{list-style-type:none}ol.lst-kix_rdjypv61h1og-0.start{counter-reset:lst-ctn-kix_rdjypv61h1og-0 0}ol.lst-kix_rdjypv61h1og-5{list-style-type:none}ol.lst-kix_rdjypv61h1og-4{list-style-type:none}ol.lst-kix_rdjypv61h1og-5.start{counter-reset:lst-ctn-kix_rdjypv61h1og-5 0}ol.lst-kix_rdjypv61h1og-8{list-style-type:none}.lst-kix_rdjypv61h1og-8 > li{counter-increment:lst-ctn-kix_rdjypv61h1og-8}.lst-kix_rdjypv61h1og-0 > li:before{content:"" counter(lst-ctn-kix_rdjypv61h1og-0,decimal) ". "}.lst-kix_rdjypv61h1og-1 > li:before{content:"" counter(lst-ctn-kix_rdjypv61h1og-1,lower-latin) ". "}ol.lst-kix_rdjypv61h1og-7.start{counter-reset:lst-ctn-kix_rdjypv61h1og-7 0}ol.lst-kix_rdjypv61h1og-2.start{counter-reset:lst-ctn-kix_rdjypv61h1og-2 0}ol.lst-kix_rdjypv61h1og-3.start{counter-reset:lst-ctn-kix_rdjypv61h1og-3 0}.lst-kix_rdjypv61h1og-5 > li{counter-increment:lst-ctn-kix_rdjypv61h1og-5}ol.lst-kix_rdjypv61h1og-8.start{counter-reset:lst-ctn-kix_rdjypv61h1og-8 0}ol.lst-kix_rdjypv61h1og-3{list-style-type:none}ol.lst-kix_rdjypv61h1og-2{list-style-type:none}.lst-kix_rdjypv61h1og-2 > li{counter-increment:lst-ctn-kix_rdjypv61h1og-2}.lst-kix_rdjypv61h1og-7 > li{counter-increment:lst-ctn-kix_rdjypv61h1og-7}ol.lst-kix_rdjypv61h1og-1{list-style-type:none}ol.lst-kix_rdjypv61h1og-0{list-style-type:none}ol.lst-kix_rdjypv61h1og-6.start{counter-reset:lst-ctn-kix_rdjypv61h1og-6 0}ol.lst-kix_rdjypv61h1og-1.start{counter-reset:lst-ctn-kix_rdjypv61h1og-1 0}.lst-kix_rdjypv61h1og-8 > li:before{content:"" counter(lst-ctn-kix_rdjypv61h1og-8,lower-roman) ". "}.lst-kix_rdjypv61h1og-3 > li{counter-increment:lst-ctn-kix_rdjypv61h1og-3}.lst-kix_rdjypv61h1og-6 > li{counter-increment:lst-ctn-kix_rdjypv61h1og-6}

Lecture 1 Don't forget about the age old question of Describe the 4 branches of chemistry.

Consider the equation Ax=b, in which a,b ∈ R

Case IWe also discuss several other topics like What is the field that looks at interactions among human systems and those found in nature?

When A x x=b/AIf you want to learn more check out What is The Wax argument?

Solution exists is unique

Case II

When A=0, we cant just divide by A

Observe that if b=0 as well, thenWe also discuss several other topics like log3 1 81

(0)x=0, so x can be every real value

                Existence of solutions but there is no uniquenessIf you want to learn more check out What is the global definition of accounting?

Case III

        When A=0 and b0, thenIf you want to learn more check out What is a spyware?

        (0)x=b, so therefore we have no solution to the problem

To generalize the version of Ax=B, we are instead of going to look for real number take x to be a real value vector.

Intuition: vectors in

Here we have . they name the same displacement and are equal

        Denote vectors using lower case letter with either “hook” or boldface.

Since we can identify a vector with a point in the plane (a,b) by starting the vector of the origin, if with (a,b) then        

                        [a]

                              [b]

Generalization - vectors in IRN

        

RN =         {[a1]                                }

        {[a2]L a;        }

        {[:  ]                                }

        {[an]                                }

When are two vectors equal to each other?

Def: let

We say if and only if

Each i=1,2..n.

Note:  

Example:         [1]

                [1]  [1]

                [0]     [1]        

Operations of vectors

Case I. Addition of vectors

Let

Example

        

=        [u1]                [v1]

        | : |                | : |

        [   ]                [vn]

        [u1+v1]

                |   :    |

                |   :    |

                [un+vn]

Case II scalar multiplication

Let         [u1 ]

                | :  |        = IRn

                [un ]                

Let c

Scalar multiplication

                [cu1 ]

                | :    |        

                [cun ]        

For c > 1, then…

        

For c(0,1) then…

        

For c < 0, then…

        Cut points in the opposite direction of , factor still applied

Subtraction of vectors

Linear combination:

Let

Let

        Linear combination of

                                        

Proposition I: let c,d

Then

1. Vector addition is commutative

()

1. 
2.         [0]

                        |0|        

                        |: |

                        [0]        

1. 
2. 
3. 
4. 
5. 

References: