Samenvatting
Cheatsheet and summary maths 2 conversion class
- Vak
- Maths2
- Instelling
- Vrije Universiteit Amsterdam (VU)
Cheatsheet for exam and summary maths 2 conversion class.
[Meer zien]Voorbeeld 1 van de 2 pagina's
Voorbeeld 1 van de 2 pagina's
In winkelwagengesponsord bericht van onze partner
Voorbeeld van de inhoud
Intro §1Gaussian elimination = row echelon form Exercise 95: Exercise 99: Solve for x & y, and
Example 46 Gauss-Jordan elimination = reduced row-
write column format & solve
echelon form
graphically.
Remark The solution to Take-home message: a “solution” may be
the linear equation ax = b an empty set, a single number, multiple
is
(countably many) numbers, or a set of Solution:
infinitely many numbers. In other words, we
generalize the concept of a solution to a set
a)echelon form of this sys eq is:
of objects that is not necessarily countable.
Exercise 100:
Solution:
Example 45 The solution set of this system of b) sol u,v,w are: u =3, v = -2 & w = 1
§§
equations is the intersection of The solution set of these two equations
c) suppose u = lnx , v =lny, w = lnz,
the two individual solution sets: then x = eu, y = ev, z = ew.
seen in figure above.
b) single solution exist = NO Matrix notation §3:
The intersection of the two solution sets
Calculate intersection by substituting the c) RHS of 3 eq = 0? = YES
1
equation is the tuple (x,y) = (2, /2) d) sys eq in lineair combination of 2 column
x = 3 in x = 5−2y. Obtaining y = 1. vectors:
à read in graph.
à So, whereas y was
. à
Then we obtain:
a free variable (i.e. y ∈ R) in the individual solution sets, 1 1 1 Column vectors
there is only a single valid value for y in the 2×2−3× 2 = 4−1 2 = 2 /2
intersection of the two sets. The solution set is a set 1 e) Non-zero choice of RHS that allows the three
with a single tuple: {(3, 1)}. Geometrically, this 2+4× /2= 2+2 = 4
lines to intersect at same point:
corresponds to one point in the plane. The à so, this solution is correct
intersection of solution sets is represented
geometrically by the intersection of the lines that Obtaining solution with algebraic X=2 & y=3, gives col vector
mapped the solution sets of the individual equations, manipulations: format:
as shown in fig. 30. So, we see a consistent 2nd eq à x = 4 − 4y
Substitution this for x : Example:
correspondence between the algebraic, set-theoretic, . .
Examples matrix multiplication
and geometric objects. 𝑥 + 𝑦 + 3𝑧 = 12. 𝑥 + 𝑦 + 3𝑧 = 12 1) exercise 97
2𝑥 + 2𝑦 + 𝑧 = 9 𝑅2
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&⃗
= 𝑅2 − 2𝑅1 −5z = −15 /////////////////////⃗
𝑅3 = 𝑅3 − 𝑅1
Graphically solution
𝑥− 𝑦+ 𝑧=2 x−y+ z= 2
.
𝑥 + 𝑦 + 3𝑧 = 12 1𝑥 + 𝑦 + 3𝑧 = 12
Substitution this for y in 2nd eq: −5z = −15 /////////////⃑
𝑅2 ⟺ 𝑅3 − 2𝑦 − 2𝑧 = −10
x = 4−2 = 2. −2𝑦 − 2𝑧 = −10 − 5z = −15 Solution:
Exercise 88: Solve w/ Gaussian el, Solution set:
determine unique solution or infinite eq3: z = 3
Exercise 83: Solve graphically and algebraically solution set. fill in eq3 in eq2: –2y – (2*3) = –10, met y = 2
fill in eq3 & eq 2 in eq 1: 1x + 2 + (3*3)= 12, met x = 3 .
So, solution is: (x,y,z) = (1,2,3). 2) exercise 96
.
Same but with matrix:
.
Solution:
Solution 1:
à then write equations and get the solution as seen above.
Remark A system of linear equations with n
variables and m equations:
• has at least n −m free variables.
Exercise 90 (same as 88): • after Gaussian elimination has at most m pivot
elements. Solution 2:
Algebraic solution: from the first equation we Nullspace of a full column rank
• has n−#pivots free variables in its solution set.
derive y = x. Substitution in the second equation
Exercise 93: matrix:
yields 3x = 6, or, x = 2. Since y = x, also y = 2
1 When it is a square matrix:
equal # of equations and
Exercise 87: Solve graphically Solution: variables. Nullspace is equal
and give solution set. to the null vector 0.
Solution set: 114 c) Find the conservation relations 2 When it is a non square
Solution:
by performing this gaussian elimination matrix: # of rows must be
1st eq = 2nd eq (if both sides larger then # of columns,
* by -2) Last eq gives z = 7,5. Subst this in 2nd otherwise matrix could not be
full column rank. So, no free
So: 2nd eq does not add eq gives:
“extra info”, so solution set variables & null space will be
Subst this in 1rst eq gives:
can by specified using eq1 the null vector, also in this
only. à gives z=4, w =3, v = 2 & u =1. Last row LHS =0, same as vector(ȧ , ḃ, ċ ) case.
Dit zijn jouw voordelen als je samenvattingen koopt bij Stuvia:
Bewezen kwaliteit door reviews
Studenten hebben al meer dan 850.000 samenvattingen beoordeeld. Zo weet jij zeker dat je de beste keuze maakt!
In een paar klikken geregeld
Geen gedoe — betaal gewoon eenmalig met iDeal, creditcard of je Stuvia-tegoed en je bent klaar. Geen abonnement nodig.
Direct to-the-point
Studenten maken samenvattingen voor studenten. Dat betekent: actuele inhoud waar jij écht wat aan hebt. Geen overbodige details!
Veelgestelde vragen
Wat krijg ik als ik dit document koop?
Je krijgt een PDF, die direct beschikbaar is na je aankoop. Het gekochte document is altijd, overal en oneindig toegankelijk via je profiel.
Tevredenheidsgarantie: hoe werkt dat?
Onze tevredenheidsgarantie zorgt ervoor dat je altijd een studiedocument vindt dat goed bij je past. Je vult een formulier in en onze klantenservice regelt de rest.
Van wie koop ik deze samenvatting?
Stuvia is een marktplaats, je koop dit document dus niet van ons, maar van verkoper arunthaskanagasabai. Stuvia faciliteert de betaling aan de verkoper.
Zit ik meteen vast aan een abonnement?
Nee, je koopt alleen deze samenvatting voor €15,89. Je zit daarna nergens aan vast.
Is Stuvia te vertrouwen?
4,6 sterren op Google & Trustpilot (+1000 reviews)
Afgelopen 30 dagen zijn er 69411 samenvattingen verkocht
Opgericht in 2010, al 15 jaar dé plek om samenvattingen te kopen