Linear Tranformation: Find the kernel of T

[MozAngeles]

Homework Statement

Let T: P₄--->P₃ be a linear transformation given by T(p)=p'. What is the kernel of T?

Homework Equations

The Attempt at a Solution

T(a₀+a₁+a₂x²+a₃x³+a₄x⁴)=a₁+2a₂x+3a₃x²+4a₄x³

Ker(T)= { T(p)=0}

so, a₁+2a₂x+3a₃x²+4a₄x³=0
then a₁=2a₂x+3a₃x²+4a₄x³

Ker(T)= { (-2,1,0,0), (-3,0,1,0), (-4,0,0,1)}

I solved this based off an example from class. But when I checked the dim[Ker(T)]= 3 and the dim[Rng(T)]=4 since my Rng(T)= {1,x,x²,x³} and dim[P₄]=5
using general rank nullity theorem I have 7=5, which doesn't make sense. So I'm wondering where I went wrong.

Thank you for your help.

 

[Samy_A]

Homework Statement

Let T: P₄--->P₃ be a linear transformation given by T(p)=p'. What is the kernel of T?

Homework Equations

The Attempt at a Solution

T(a₀+a₁+a₂x²+a₃x³+a₄x⁴)=a₁+2a₂x+3a₃x²+4a₄x³

Ker(T)= { T(p)=0}

so, a₁+2a₂x+3a₃x²+4a₄x³=0
then a₁=2a₂x+3a₃x²+4a₄x³

Ker(T)= { (-2,1,0,0), (-3,0,1,0), (-4,0,0,1)}

I solved this based off an example from class. But when I checked the dim[Ker(T)]= 3 and the dim[Rng(T)]=4 since my Rng(T)= {1,x,x²,x³} and dim[P₄]=5
using general rank nullity theorem I have 7=5, which doesn't make sense. So I'm wondering where I went wrong.

Thank you for your help.

Ker(T) contains all polynomials in P₄ whose derivative is 0.

Look at your answer: Ker(T)= { (-2,1,0,0), (-3,0,1,0), (-4,0,0,1)}.
A polynomial in P₄ is defined by 5 coefficients. What polynomials in P₄ do these elements consisting of 4 coefficients represent?

 

[MozAngeles]

Ker(T) contains all polynomials in P₄ whose derivative is 0.

Look at your answer: Ker(T)= { (-2,1,0,0), (-3,0,1,0), (-4,0,0,1)}.
A polynomial in P₄ is defined by 5 coefficients. What polynomials in P₄ do these elements consisting of 4 coefficients represent?

I realize where i went wrong and found my kernel to be (0,0,0,0,1) , which then the rest makes sense

 

[Samy_A]

I realize where i went wrong and found my kernel to be (0,0,0,0,1) , which then the rest makes sense

That is correct, sort of.
A good examinator would ask what (0,0,0,0,1) actually represents in P₄. In other words, what set of basis vectors are you using? Without that information, (0,0,0,0,1) is meaningless as answer.
Moreover, Ker(T) is not a vector, but a subspace.

 

[MozAngeles]

That is correct, sort of.
A good examinator would ask what (0,0,0,0,1) actually represents in P₄. In other words, what set of basis vectors are you using? Without that information, (0,0,0,0,1) is meaningless as answer.
Moreover, Ker(T) is not a vector, but a subspace.

If my thinking is correct...
{(0,0,0,0,1)} represents the set of vectors that are an element of P₄ such that the linear transformation of these vectors, in P4, is a homogeneous equation

 

[Samy_A]

If my thinking is correct...
{(0,0,0,0,1)} represents the set of vectors that are an element of P₄ such that the linear transformation of these vectors, in P4, is a homogeneous equation

What linear transformation? How does a homogeneous equation come into play?

Back to basics: Ker(T) is a subspace of P₄.
Your answer should say what polynomials are in that subspace. If you want to represent the elements of P₄ in some basis, you have to make clear what basis you are using.
You seem to know what the answer is, but you don't convey it clearly at all.

 

[MozAngeles]

What linear transformation? How does a homogeneous equation come into play?

Back to basics: Ker(T) is a subspace of P₄.
Your answer should say what polynomials are in that subspace. If you want to represent the elements of P₄ in some basis, you have to make clear what basis you are using.
You seem to know what the answer is, but you don't convey it clearly at all.

i was using the definition from by book for kernel space. Being that the initial prob stated it was a linear transformation I used the definition: Ker(T)= {v element of V: T(v)=0) which in words is If T: V--> W is any linear transformation, there is an associated homogeneous linear vector equation T(v)=0. In this case Ker(T)= {(a4,a3,a2,a1,a0):(0,0,0,0,a0}

 

[Mark44]

i was using the definition from by book for kernel space. Being that the initial prob stated it was a linear transformation I used the definition: Ker(T)= {v element of V: T(v)=0) which in words is If T: V--> W is any linear transformation, there is an associated homogeneous linear vector equation T(v)=0. In this case Ker(T)= {(a4,a3,a2,a1,a0):(0,0,0,0,a0}

What Samy_S was asking was, what polynomials are in the kernel? P₄ is a (function) space of polynomials.

Writing an answer of (0, 0, 0, 0, a0) is meaningless if we don't know what the basis is that you're using. In particular, we don't know what polynomials are represented by this vector.

 

[blue_leaf77]

i was using the definition from by book for kernel space. Being that the initial prob stated it was a linear transformation I used the definition: Ker(T)= {v element of V: T(v)=0) which in words is If T: V--> W is any linear transformation, there is an associated homogeneous linear vector equation T(v)=0. In this case Ker(T)= {(a4,a3,a2,a1,a0):(0,0,0,0,a0}

The helpers basically wanted to know how are the entries in your matrix representation are ordered? When you write $(a,b,c,d,e)$, are your ordering them according to $\textrm{coeff}(x^0,x^1,x^2,x^3,x^4)$ or $\textrm{coeff}(x^4,x^3,x^2,x^1,x^0)$ or $\textrm{coeff}(x^3,x^1,x^2,x^4,x^0)$ etc?