First Year Mathematics B (Integration, Series, Discrete Maths & Modelling)

[leehuan]

What other method do you have in mind? (Induction helps takes care of it nicely if you want to prove it straight from the axioms.)

For some reason I kept thinking what was the problem with doing it component wise...

In that case, how could you prove this then?


As a reference, they proved like this


Also gonna take a stab at the axioms they used...
1. Existence of 0
2. Scaling factor of 1
3. arithmetic (or just existence of 0 in field of scalars)
4. Scalar distributive law
5. Scaling factor of 1

[InteGrand]

For some reason I kept thinking what was the problem with doing it component wise...

In that case, how could you prove this then?


As a reference, they proved like this


Also gonna take a stab at the axioms they used...
1. Existence of 0
2. Scaling factor of 1
3. arithmetic (or just existence of 0 in field of scalars)
4. Scalar distributive law
5. Scaling factor of 1

We can't just do these component-wise because they are results about arbitrary vector spaces (not necessarily ones where the vectors are n-tuples).

[RenegadeMx]

For some reason I kept thinking what was the problem with doing it component wise...

In that case, how could you prove this then?


As a reference, they proved like this


Also gonna take a stab at the axioms they used...
1. Existence of 0
2. Scaling factor of 1
3. arithmetic (or just existence of 0 in field of scalars)
4. Scalar distributive law
5. Scaling factor of 1

man i feel sorry if they expect u to properly remember the axioms, for us we just had to use 2 or 3.

[seanieg89]

They really aren't hard to remember.

[RenegadeMx]

They really aren't hard to remember.

annoying as fuck tho

[leehuan]

The axioms are just annoying.

Actually knowing when and how to use them like there though is ...

[Flop21]

S = {x is an element of R^4 | (x1)-5(x3) = 2(x4)}


Find one non-zero element in S.


Do you just sub in random values for x1,x3,x4 (also where's x2)?

[RenegadeMx]

S = {x is an element of R^4 | (x1)-5(x3) = 2(x4)}


Find one non-zero element in S.


Do you just sub in random values for x1,x3,x4 (also where's x2)?

x=(2,0,0,1) x2 can be whatever u want doesnt affect the eqn

[leehuan]

Don't mind if this question isn't finished off for me but can someone please set it up? Finding it hard to set everything out neatly here.

[HeroicPandas]

Don't mind if this question isn't finished off for me but can someone please set it up? Finding it hard to set everything out neatly here.

Is there missing detail like Ax = 0 or Ax is an element of ___

[leehuan]

Is there missing detail like Ax = 0 or Ax is an element of ___

Nope, and I know it can't be Ax=0 because that's a system of linear equations.

Pretty sure by assumption Ax is an element of R3 if the whole set is a subspace of R3

[HeroicPandas]

Nope, and I know it can't be Ax=0 because that's a system of linear equations.

Pretty sure by assumption Ax is an element of R3 if the whole set is a subspace of R3

Woops, I did a mistake in dimension and thought that Ax is still 3x5 matrix

[RenegadeMx]

Don't mind if this question isn't finished off for me but can someone please set it up? Finding it hard to set everything out neatly here.

i forgot the proper proof for these

U is subset of r3 which is known v.s <- not sure if u need to include this for first year, but im used to writing it anyway..

0 vector is obvious
Ax+Ay=A(x+y) y element of U etc i.e A(0)+A(0)=A(0+0) = 0
A*(Landa)(x)=(Landa) Ax landa is real, can be complex if u want A*Landa(0)-Landa A(0) =0

[InteGrand]

Don't mind if this question isn't finished off for me but can someone please set it up? Finding it hard to set everything out neatly here.

[InteGrand]

In other words, to show a set S is closed under addition, it means we have to show that if a and b are arbitrary elements of S, then so is a+b.

To show the set U is closed under scalar multiplication, it means we have to show that if c is an arbitrary scalar and v an arbitrary vector from U, then cv is also in U.

[RenegadeMx]

heres another one where u need to think more carefully compared to the easy stuff like 3x1+2x3=0 (if u want the practice)

[Flop21]

For this polynomial question, I get the same matrix but R1 and R3 swapped. Is this still okay, or is my method wrong?

[InteGrand]

For this polynomial question, I get the same matrix but R1 and R3 swapped. Is this still okay, or is my method wrong?

As long as you have the same rows, you'll end up at the same answer (because then you're starting off with the same set of simultaneous equations, just written in a different order, which clearly won't affect the solutions).

[leehuan]

heres another one where u need to think more carefully compared to the easy stuff like 3x1+2x3=0 (if u want the practice)

I'll go for it if I get the time

[leehuan]

^Feelsbad delaying the exercise

Anyway



Is this just because the zero polynomial can't be in S?

[RenegadeMx]

^Feelsbad delaying the exercise

Anyway



Is this just because the zero polynomial can't be in S?

yes

[RenegadeMx]

to prove its not a v.s in general u either show 0 isnt there (easiest) or that it doesnt satisfy addition/scalar multiplication

only need to show 1 of those conditions as well

[leehuan]

yes

Lol sweet.
_______________________

Even if there IS a typo in the notation, this question is actually confusing me to the maximum...



This was the answer I got to part (ii)

[leehuan]

to prove its not a v.s in general u either show 0 isnt there (easiest) or that it doesnt satisfy addition/scalar multiplication

only need to show 1 of those conditions as well

Yeah sweet, that reaffirms what I know

[Paradoxica]

Lol sweet.
_______________________

Even if there IS a typo in the notation, this question is actually confusing me to the maximum...



This was the answer I got to part (ii)

You forgot that α⁵=1