Linear Algebra Q (2 Viewers)

[zeebobDD]

 

[Carrotsticks]

A few Q's (don't want to give the answer straight to you).

1. Do you know what the definition of a subspace is and what the properties of a subspace are?

2. Do you know (geometrically) what S is representing? Hint: Think of Extension 2 Mathematics...

 

[zeebobDD]

Well i get 1. , i've done it by choosing two general equations as F(x) and G(x) which are both in space S

then using those proved axiom A1 and S1 , (f+g)(1)=f(1)+g(1)=0 and same way for the derivative, but i'm not sure if it's right or not

and i can't seem to figure out what this represents geometrically :S

 

[Carrotsticks]

Well i get 1. , i've done it by choosing two general equations as F(x) and G(x) which are both in space S

then using those proved axiom A1 and S1 , (f+g)(1)=f(1)+g(1)=0 and same way for the derivative, but i'm not sure if it's right or not

and i can't seem to figure out what this represents geometrically :S

Since P(1) = 0, it means P has a root at x=1. However, since P'(1) = 0 too, it means that the root at x=1 is a double root.

So S is the set of all polynomials (of degree at most three) satisfying P(1)=0 and P'(1)=0. For a quadratic and cubic, this means it has a double root at x=1.

By A1 and S1, I presume you mean the addition and scalar properties respectively?

I'll give you the 'English translation' to make things a bit easier for you. Seems like you're having difficulty understand what the question is asking. You already know that to prove that S is a subspace of P_3, you have to prove 3 properties.

Addition: If you add any two such polynomials (satisfying the condition that P(1)=P'(1)=0), you should also have a polynomial that still satisfies that condition.

Scalar: If you multiply a polynomial in S by a scalar, it still satisfies the above condition.

Zero Vector: The zero vector (in this case the zero polynomial) is in S.

To prove S is a subspace of P_3, you need to show that S satisfies all these three properties. Give it a try now!

 

[zeebobDD]

Right got it, i've done the solution but my method uses two general cubic equations,ax^3+bx^2+cx+d.. etc

is this allowed?

 

[Carrotsticks]

Right got it, i've done the solution but my method uses two general cubic equations,ax^3+bx^2+cx+d.. etc

is this allowed?

Can I see your working? That doesn't seem right because the cubics you're dealing with are not general at all. They must satisfy P(1)=P'(1)=0 so they will be in the general form P(x)=(ax+b)*(x-1)^2

 

[zeebobDD]

Hmm alrite my equations must be wrong, i'll try it with your general form ones , post it tonight

 

[zeebobDD]

okay done it for S1 and A1, it satisfies the equation , how do i express that it's non empty formally, i can't just write it?

 

[seanieg89]

Well the zero polynomial is clearly in S, so it is non-empty.