Linear Tranformation: Find the kernel of T

Homework Statement

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

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.

Answers and Replies

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Samy_A
Science Advisor
Homework Helper

Homework Statement

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

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?

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
Science Advisor
Homework Helper
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.

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
Science Advisor
Homework Helper
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.

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
Mentor
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
Science Advisor
Homework Helper
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?