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Chapter 01: System of linear equation and matrix V1 — Transcript

by Institute of Digital Technology - IDT · 4,064 words · 677 segments · language en · Watch on YouTube

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0:00foreign
0:00[Music]
0:07today I will introduce you about linear
0:11zebra
0:12so in linear algebra we have more about
0:15relation in here
0:17the first of our relation in linear
0:20algebra I talk about the system of
0:22linear equation and Matrix Matrix so the
0:25first
0:26today I will introduce you about the
0:30system of linear equation so what is
0:32linear equation before you know about
0:34the system or linear equation you should
0:36we know about the linear equation
0:38I think you know me already uh
0:42the the old generation but you are the
0:46new generation of uh ninth generation I
0:49think
0:51maybe you know my name my name is and I
0:54do that in here and I am a lecturer at
0:57cadt also uh mathematic lecture
1:02so now I will uh introduce you or I will
1:06teach you about deletion of system of
1:09linear equation
1:10so now uh linear equation the first
1:14point we talk about the system of linear
1:17equation and Matrix
1:19and the next
1:21we talk about the ghost elimination so
1:24what is the ghost elimination
1:27and the next
1:30we talk about the linear equation
1:33we talk about the linear system equation
1:37we talk about the elementary row
1:39operation
1:41we talk about the ghost scene
1:43elimination so our main point in this
1:46lesson we talk about the gaussian
1:48elimination
1:49how can you solve the system of linear
1:52equation by gaussian elimination
1:55and this is our expect our output
1:59after you finish this lesson you can get
2:02knowledge of system of linear equation
2:05understanding and soul sum of system of
2:09linear equation
2:11and create to solve the system create to
2:14solar system it means that you can
2:16make your xxi you can make more exercise
2:20by yourself and to solve in your group
2:22or in virtually for your uh
2:27learning
2:28so our learning outcome objective in
2:32distillation you will be able to study
2:34about ocean elimination
2:36linear equation
2:38system of linear equation
2:40Elementary row operation and gaussian
2:43elimination so the first we talk about
2:46the system of equation
2:48H geometry
2:51our outcome related to a type of
2:54solution of a system of two or three
2:57variables to the intersection of line in
3:01a plane the intersection applied in this
3:05three space
3:07it means that we talk about in our space
3:10and three dimension you may remember
3:13this is uh the the system of
3:18this is not system of linear equation
3:20this is only uh
3:23linear equation so you should be you
3:26know linear equation and sometimes you
3:28call this is the line equation also
3:31so linear equation is the equation that
3:34have
3:35uh two water board or three bar wall but
3:39have only
3:40power one the degree is one degree of
3:44our variable is one so this is we call
3:47linear equation
3:48so equation
3:50uh we talk about we think about the
3:54right hand side and the left hand side
3:56so right hand side equal to left hand
3:58side and the vertical is power one so we
4:01call this this linear equation like I
4:04cycle for you on 2x plus 3y equal to six
4:08so this is recall about the uh
4:13linear equation so you can solve linear
4:17equation from Grade 9 grade 10 grade 11
4:21and grade 12 also so how can you solve
4:24about this uh linear equation when you
4:27want to find X
4:29you should be give the value of y and
4:33when you want to find y you will give
4:34the value of x also or example if you
4:38want to X Plus y
4:41example 2x plus y
4:44to X Plus
4:463y equal to 6 so you can give this one
4:50example you give Y is equal to zero and
4:54X is equal to three so you got three
4:58multiplied by two multiplied by three
5:01and plus three multiplied by zero is
5:04equal to six so this is true
5:07or sometimes you can take uh
5:112X and write 2x plus 3y equal to 6 4 to
5:176 and you can give this one is like a
5:21uh
5:22I can give this one is uh zero two
5:27multiplied by zero and plus 3 multiplied
5:31by two is equal to six also
5:34so you have more why this is you have
5:36more about the solution in some time in
5:40uh integer number number and sometime in
5:43real number also example is this I give
5:46you about this value this is only about
5:50uh
5:51integer number but in real number
5:54example I give two to X
5:58plus 3y
6:00equal to 6 equal to 6. so it means that
6:04in here
6:06I can
6:07give more about the value of X and Y X
6:12and Y example I give uh
6:16two
6:18multiplied by 1 and plus 3 multiplied by
6:23uh
6:28like six multiplied by 6 this is equal
6:32to uh
6:3420.
6:36equal to 20
6:38but our example this is only equal to 6.
6:42so how can you do this one to B6 so this
6:46is 18.
6:48this is 18. so x 10 plus 2 equal to 6 it
6:53means 18 plus 2
6:56become too sick so extinct how can you
6:58divide about extinct in here
7:00you know so there are many of our
7:04solution in uh
7:07linear equation
7:09because linear equation this is the line
7:12example I have this
7:14graph
7:16y prime or Y and X Prime o x so example
7:21this is a line
7:23this is a line so the intersection
7:25between X and Y
7:28between X and Y between X and Y between
7:32X and Y
7:33uh between X and Y
7:36the intersection between X and Y this is
7:38about like my point over here
7:46this is the solution of our equation
7:50so please remember that linear equation
7:54have more solution in here
7:56and the next point I will show you how
7:59to solve a linear equation in
8:03integer number
8:04integer number or in real number or in
8:08natural number how can you solve this
8:10equation and I will tell you more about
8:13more deeply about this uh solving of
8:17linear equation in the next point
8:20and let's start when we look to the
8:23intersection of the line
8:25so this is the first graph which I draw
8:29for you this one it means that we have
8:32about the line
8:34this is line one and this is line line
8:37two so this point this is the
8:40intersection of line one and line two
8:43you think line one and line two have
8:46only one intersection here in the
8:49section here so line one and line two it
8:52means that
8:54uh this is about the system of linear
8:57equation example this is line one
9:01and this is line two
9:04so line one in line two we have
9:06intersection one like a
9:09X row and y y zero so our system of
9:13linear equation have one solution
9:16so I I right here this is have only one
9:20solo solution because the intersection
9:22of line one and line two have only one
9:26point if have only one point it means
9:29that this equation or the system linear
9:32equation have only one slow solution and
9:36now you go to the next
9:38this is the next graph
9:41in a graph the next graph you see about
9:44the line one I
9:46let the red line is line one and the
9:49blue is the line two
9:51so you think line one and line two this
9:54is not intersection just on
9:57parallelization Parallel so a parallel
10:00so line one and line two cannot be
10:03intersection
10:04if cannot be except intersection line
10:07one and line two it does not meet each
10:10other
10:11when Line 1 and line two is parallel or
10:14not meet each other so our system of
10:17linear equation have no solution
10:19example if I get X plus 3
10:23plus 2y equal to Y and X
10:29Plus
10:30uh
10:33minus minus I put minus 2
10:37y equal to 9. this is on the example
10:42so if I draw about
10:45the graph
10:47example this is the graph and this is
10:50the line of
10:51this one and this is two so one and two
10:55so one and two cannot intersection so
10:58our system of linear equation cannot be
11:01cannot be have solution or we can say
11:05this is no true no solution and now you
11:09go to the graph three in graph three
11:12it means that you have you see only one
11:15line so if you draw example 2X
11:22plus 3y this is equal to
11:25uh six this is on the example I write
11:28for x and plus 6 y equal to 12.
11:33so if we know about the system of linear
11:37equation it means that the first
11:40the first equation
11:42is 2x plus 3y equals 6. and the second
11:48equation is 4X plus 6y equal to 12.
11:53equal to 12 in here
11:55so you think about the
11:58equation one equation one is 2X
12:02plus 3y equal to 6.
12:05and equation two
12:07equation two is for x
12:10plus six y equal to 12 and you can
12:15divide by two you can divide by two and
12:18divide by 2 you have 2X
12:20plus 3y equal to 6. so equation two and
12:25one
12:26equation 2 and 1 is the same
12:29when equation 2 and 1 is equal each
12:33other it means that this line is redone
12:36or this line is on the sample position
12:39like you see about this line the red
12:42line this is only one one line so if
12:45only one line you want to find about the
12:48solution it means that intersection
12:51between X and Y
12:53intersection between X and Y
12:56intersection between X and Y
12:58intersection between y this is
13:02the solution of this equation so have
13:05more
13:05more about
13:07intersection between x and x and y
13:11so
13:12in this graph in this graph it means
13:16that our solution
13:18have Infinity solution or many slow
13:21solution in here so you should be know
13:24about in our system we have
13:29three case
13:30one case is only one solution no
13:33solution and infinity minus Lu solution
13:38and for the next
13:41for the next we talk about the uh
13:45the system of equation like X Plus y
13:49equal 1 and 2x plus 2y equal
13:532. so you you change you can check about
13:56this equation X Plus y equals one and
14:00two X plus two y equal to so this one
14:03when you divide by two you got X
14:06plus y equal to one so between equation
14:091 and equation two are equal
14:12so when equal it means that you have
14:15many slow many solution if you write X
14:19Plus y equals 1 and 2x plus 2y equal
14:24equal 2. so this is
14:27this system or linear equation have many
14:29slow solution because the graph the
14:32graph is redone or in the same policy
14:36the same position
14:40and the next for example a graphical
14:43solution so you can solve the system or
14:46linear equation by a graphical solution
14:49use a graph to find solution
14:52the follow of system of equity equation
14:57example you have equation one is X Plus
15:01y equal 3 and equation two is X Y minus
15:06X is equal to Pi
15:08so in this case done don't be confused
15:12about this point
15:14uh in
15:15our country when the use about the
15:18system
15:21system equation
15:26equation they write in a curly bracket
15:29like X
15:31plus y equals 3 and Y minus eight equal
15:37to equal to 5. so this is right in our
15:41country you write in curry black blanket
15:46but
15:47other country or on glow section or
15:51something in Singapore in UK they write
15:55about the system of linear equation like
15:58this one
15:59yeah I like this one they do not need to
16:02juice about the curry curry blanket
16:06so please remember remember about this
16:09point
16:10and the next the next reveal
16:14solution through graphing about the
16:16equation and identify the
16:20resist identify the craft point
16:22intersection we can find solution
16:26in here so what is the
16:33the line that represents who the
16:35equation one and which line is
16:38represented to the equation two this is
16:41equation one and this is equation two
16:43so equation one is when we can we can
16:47write into this here Y is equal to
16:50Negative X plus 3 and this is uh y equal
16:55to X plus pi so you can draw about the
17:00line or the graph of this equation like
17:04you give the value of X and Y
17:07so when you get 8 0 y E3 and y0 and X is
17:13equal to 3 or so
17:15and if this here is like X and Y
17:18again y you can
17:23you can give like uh x0 and 5 y 0 this
17:28is the negative of of I
17:31so in here you can guess which one is
17:34the equation 1 and which line is the
17:36equation two the green one
17:39and the the blue one so the green one it
17:43means that
17:45the green one is equation one like
17:49X
17:51plus y equal to 3.
17:53and the blue one is y minus x equal to 2
17:585.
17:59but you see this equation
18:02this line the green line and the blue
18:05light blue line intersection at this
18:08point it means that when you
18:11orthogonal or projects
18:14project orthogonal on x-axis you got
18:17-1
18:18and project on
18:21y-axis you got this is four
18:24so the intersection between red between
18:28blue line and green line is at the point
18:31X and Y that X is equal to -1 and y
18:36equal to 2 4. so
18:39our system of unit equation x
18:42plus y equal 3 and Y
18:46minus x equal to 5 we got the value is X
18:51and Y
18:53that x equal to minus 1 and y
18:56equal to 2 4 this is our slow solution
19:02so this is only my explanation about the
19:05system of linear equation in
19:07our Cartesian plane like only in plan X
19:12and Y
19:13so now
19:15we check for the intersection Point
19:18remember the must-have either one
19:21solution Infinity solution or no
19:24solution at at all so which one
19:27the following all the graphs so that our
19:31uh intersection of our
19:35equation one and equation two we have
19:37only one one point only one point so the
19:42system of linear equation have only one
19:45slow solution that x equals to minus one
19:49and y equal to to 4.
19:53and for the next
19:56we will talk about the system of linear
19:59equation also so now you know already
20:02when you think about the system only
20:05equation you can solve the linear
20:07equation by our our example you have
20:11this graph and you have this line
20:14and this is the intersection of
20:18line Bond and line two so like this one
20:21is y0 and this is x0 so
20:25our system or linear equation have like
20:28X and Y
20:29equal to X rho N Y zero this is about
20:34the solution of our query equation
20:38and
20:40also this is in plan
20:43also X and Y but we have three
20:47example we have equation one
20:51equation 2 and equation 3. equation
20:55three
20:56so this is I can call this is line one
21:00line two and line three also
21:03line one line two and line three you can
21:06think about
21:09the intersection of line one line two
21:11and line three
21:13so line one can be intersection with
21:18line three and line two
21:23line three and line two line one
21:26but
21:28we have
21:30different point
21:32example
21:33line one intersection with line three we
21:37have this one is x0 and Y wise row
21:42line one intersection with
21:45line two
21:46with line two we got X row but this one
21:49is y0 print is it different from y0 it
21:54is not at the same point
21:56so and line two intersection with line
21:59three
22:01line tool in the section with line 3 it
22:04means that you got about X1
22:07and this is we can call Y y1 so line one
22:12in the section we have
22:14I can write this is A1
22:17A2 and a A3 so intersection between
22:22three line it is not at the same point
22:25it's different point line one in the
22:27section line two line one into six and
22:29line three line two in the section with
22:32line one line one in the section with
22:34line three uh line two and line three it
22:37is the different place different place
22:40if different place it means that we
22:43cannot
22:44we cannot say that this equation one
22:48equation two and equation three have no
22:52solution
22:53because it is different point
22:55example line one and line two is true
22:58but line one and line three it is not
23:00true this is four
23:02it is not example it is not 2 equal to
23:07or two
23:09equal to 3 so it is not true
23:12so this is no slow no solution
23:17and for the next
23:21this graph it means that line one
23:24line one line two and line three
23:26intersection in the same point
23:29only one point like this one I can call
23:31this this a
23:33half about the Y zero
23:36and this is an x x row so it means that
23:40line one intersection with line two
23:44intersection with line three
23:47have
23:48about in here
23:51like X row and
23:54and vice row if it have x row and y0 in
23:58the same point it means that equation
24:00one equation two equation three
24:04combine each other we can call this the
24:06system of linear equation
24:08so system of linear equation in line one
24:10line two and line three it means that we
24:13have one intersection so the system have
24:16only one slow solution
24:22and consider the first picture below
24:24while three line intersect with one
24:28another there is no common point in the
24:31section where all three line meet at one
24:35point
24:36and there is no solution of three
24:39equations
24:40remember a solution of a point X and Y
24:44which certify all three equations
24:48in the case of second feature the line
24:51intersection at common point this means
24:54that there is no solution to three
24:57equation
24:58like I asked Can you about
25:02and
25:04the next we talk about the first
25:09the first thing I talk only about in
25:12plan
25:12in plan in plan like extreme o x
25:18o y prime or Y and extreme o x this is
25:24in plan
25:27this only plan but for next point I will
25:30introduce you about intersection in
25:34spice
25:36X
25:38y and z
25:40this is only o in space
25:42so in space
25:45this is only juice about the
25:48variable X and Y
25:50and this is we use x
25:53Y and Z and halves Vector I
25:58Z and K also
26:00this is only I and z
26:05so
26:06now when we talk on Space we cannot
26:10think about the
26:12intersection between line intersection
26:15plane
26:16another plan to another plan like you
26:19think on your wall or at your house the
26:23the wall you think this is one plan and
26:26your
26:27floor this is one plane intersection
26:29between your wall and your fall this is
26:33only one one line example in here if I
26:39I draw about this
26:43this plan this is a call
26:45plan p
26:47and another one like that I put
26:52plan
26:54Q
26:56so intersection between plan p and Q
27:00p and so this is one one line over here
27:03so this is the intersection of plan p
27:07and Q
27:08so now we go to our relation for next
27:12you may recall the equation in three
27:16variable sex to X plus 4y and minus 5z
27:22equal to 8 from a plane about
27:25we we were looking to intersection of
27:29line in other to identify any possible
27:32solution when graphical solving system
27:35of equation in three vertical
27:38we look
27:40we look for intersection of plan
27:43this point in the section give X Y and Z
27:46in here X Y and Z that certify all
27:51equation in system
27:53so what high of solution are possible
27:56when working with three variable
27:59consider the following picture involved
28:02to plan
28:04which are given by two equation in three
28:07variables
28:09so the first example I showed you only
28:13one
28:15uh equation in two variable but now I
28:20will show you about one equation in
28:22three variables so what happened next
28:26so this is like plan
28:28P right plan p this is like plan Q then
28:34Q so this is the intersection of p and Q
28:38this is intersection between pen p and Q
28:42so p and show this is the parameter line
28:46also in our plan or in our space I have
28:52you study already in grade 12 the
28:54intersection between plan and plan you
28:57got the parameter of the query uh
29:00equation in line
29:03so now
29:05we will go to see this is three plan
29:09three plan also
29:11we get about a new plan of X Y and Z the
29:15coordinator
29:18and this is also a new plan that you got
29:22from intersection plan
29:26this is also when you have this is
29:29intersection between a
29:34another plan
29:40now
29:42seeing how three equations in three
29:44verbal can be have no solution a unique
29:48solution intersection in the lines uh
29:51resulting in Infinity solution
29:55it is also possible that three equations
29:58graph the same plane
30:00which also lead to infinity minutes low
30:03solution
30:07like equation like X
30:10so this is informable X
30:13plus Y and plus uh
30:18minus 2z and plus 4 W equal to X
30:23with more than three variables are often
30:25called hyperplane
30:28hyperplane you may soon release that it
30:31is tricky to draw the graph of
30:34hyperplane
30:36through the toll of linear algebra we
30:40can algebraically examine this type of
30:43system which are difficult to graph
30:46in the following section we will
30:48consider this algebraic tool also
30:57so this is only xxi
31:01we talk about the graphical to find
31:03about the point X1 and y1 between uh
31:09equation 1 and equation two so when you
31:12do you can write in our curly blanket is
31:15X1 Plus
31:173y equal one and four x
31:22minus y is equal to 1. so you can
31:27multiply this by negative four
31:32you got minus 4 X Plus 3y
31:36equal 1 and 4x
31:39this is
31:41equal to -4
31:43F4 and 4x minus y equal to 1.
31:48so X that's all you got for y
31:53equal to
31:55uh this is minus 4 equal to 3.
31:59-3
32:00so you got y this is equal to minus
32:03three oh minus four is three over four
32:05you can substitute Y into y you got
32:11for x
32:13minus 3 over 4y equal to 1. so again you
32:19can find about y this is equal to
32:21something
32:22so when you do this one it is not by the
32:26equation
32:29when you do this one for our design you
32:31got zero point in here
32:34because
32:35our exercise they they want you to find
32:39about the X one and Y one
32:43which lie on the Boss line
32:45biographically
32:47so they want you to solve this
32:50system of linear equation by graph
32:54when you saw by a substitute or by other
32:57operation it is not
33:00you cannot get the score for your
33:03solving
33:04please remember about this point before
33:06you sold about the point or other
33:09equation or other exercise you should be
33:13read be carefully about hypothesis what
33:16we want you to solve about this hxi
33:19please remember about this point
33:22and this is on low only about it's
33:25excited or so
33:28and it's excite 1.1 and 0.3
33:34the top in general
33:37you have a system of K equation okay
33:40equation it means that up to you
33:44in two variable in two word ball it
33:47means that the UK
33:51bigger or small bigger or equal to two
33:54explain the geometric
33:58significance of no solution
34:01unique solution and infinity solution
34:05if you want to know about the equation
34:08the system of K equation in two variable
34:12have no solution what does mean if it
34:15has no solution unique solution and
34:19infinity solution in here
34:22so explain in geometry
34:26significant
34:28so this is a sign that we will talk in
34:30our class
34:31before
34:33you meet me in class you should be
34:35thinking about this all this exercise
34:38but
34:39you think and you do by yourself or with
34:42your group
34:44and when you meet me in the class I will
34:46ask you to solve about this problem on
34:49whiteboard or we can top each other in
34:51group in our class
34:53foreign
34:57[Music]

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