# Finding a linear transformation.

• peripatein
In summary: It gives you the equation px+q=0, which can be solved for x and q to get the desired transformation.In summary, the person is trying to find a linear transformation which satisfies T: R3->R1[x], ker T = Sp{(1,0,1), (2,-1,1)} but is having trouble understanding why they cannot find one. They understand that there is no linear transformation which satisfies T, but are not sure how to prove it. They also understand that dim(R1[x]) is 2, so there is no such transformation. They try to use a dimensional analysis to disprove that no such linear transformation exists but are not sure how to do so. They also ask
peripatein
Hi,

## Homework Statement

How may I find (or prove that there isn't) a linear transformation which satisfies T: R3->R1[x], ker T = Sp{(1,0,1), (2,-1,1)}?

## The Attempt at a Solution

I am not sure how to approach this. I understand that kerT is the group of all vectors (x,y,z) in R3 so that T(x,y,z) = 0 = Sp{(1,0,1), (2,-1,1)}. So x = alpha, y = -beta, z = alpha + beta? Hence, x,y,z=0? Hence, T has to be one-to-one? Since dim(R1[x]) is 2, does that mean that there is no such linear transformation?

I don't understand what you try to do with your alpha and beta.
Every vector which can be written in that way is part of ker T - and now? What about vectors which cannot be written like that?

peripatein said:
Since dim(R1[x]) is 2, does that mean that there is no such linear transformation?
You can use a dimensional analysis to prove that there is no surjective linear transformation (with the given ker T), but that is not relevant here.

Won't dimensional analysis show that dimIm(T) must be equal to 1? How does that help me disprove that no such linear transformation exists?

Why do you think you can disprove it at all?

dim Im(T)=1, I agree

I don't know, wrong inference from your previous post I presume. In any case, how shall I proceed?

$R_1[x]$ is the set of all polynomials of the form ax+ b for real numbers a and b (unless I am misunderstanding- I would call that R[x] without the "1"). We can represent that by the column vector $\begin{bmatrix}a \\ b \end{bmatrix}$ and so we can represent a linear transformation from $R^3$ to $R_1[x]$ by a 3 by 2 matrix:
$$\begin{bmatrix}a & b & c \\ d & e & f \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}= \begin{bmatrix}p \\ q\end{bmatrix}$$

Saying that the kernel is spanned by (1, 0, 1) and (2, -1, 1) means that
$$\begin{bmatrix}a & b & c \\ d & e & f \end{bmatrix}\begin{bmatrix}1 \\ 0 \\ 1 \end{bmatrix}= \begin{bmatrix}a+ c \\ d+ e\end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$$
and
$$\begin{bmatrix}a & b & c \\ d & e & f \end{bmatrix}\begin{bmatrix}2 \\ -1\\ 1\end{bmatrix}= \begin{bmatrix}2a- b+ c \\ 2d- e+ f\end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$$

So we must have a+ c= 0, d+ e= 0, 2a- b+ c= 0, and 2d- e+ f= 0. That gives four equations with which we can solve for four of a, b, c, d, e, and f in terms of the other two.

Would you care to explain the logic behind such approach? For instance, why did you choose a matrix whose first two elements in the first row are the elements of the column vector [a,b]?

a to f are just some constants (with arbitrary names), as this 2x3-matrix represents an arbitrary linear transformation from R^3 to R^2 (or R[x] in this special case).

Right, but it is for a reason that a and b appear in both cases, namely first two elements in the first row of the matrix AND the elements of the column vector [a,b]. I am asking why is that.
Moreover, I have a general question - are all different bases linearly dependent?

No, they are not related. As you can see, HallsofIvy replaced the names by p and q in the first equation to avoid collisions.

Okay. The result I obtained was a=b=-c and d=-e=-f/3.
How do I derive my answer from these values?

As you have no additional restrictions, you can choose two values for those parts to find a linear transformation.
a=b=-c=5, d=e=-f/3=7 for example.

Hence, how should the final answer be presented? Shouldn't it be presented in the form of a polynomial ax+b?

The final answer should give some unique way to map all vectors (g,h,i) to vectors px+q.

In that case, I don't think I understand. I thought I was to provide a general term for a transformation answering the criteria as presented in the question. Would you clarify, please?

You have to give a single linear transformation - this is a specific way to map every vector of R^3 to a polynomial: Like an equation T(g,h,i)=... which can be used for all values of g,h,i and gives a result which depends on g,h,i.

Let me see if I got it correctly. Supposing I take the values for a,b,c,d,e and f, and substitute them into the original matrix as suggested by HallsofIvy, I get a single column vector whose elements are 5x+5y-5z and 7x+7y-21z. To what end does that serve me?

## 1. What is a linear transformation?

A linear transformation is a mathematical function that maps a vector space onto itself in a linear fashion. This means that the output of the function is a linear combination of the input vectors.

## 2. What is the purpose of finding a linear transformation?

The purpose of finding a linear transformation is to understand and manipulate the relationships between different vector spaces. Linear transformations can be used in various fields such as physics, engineering, and computer science to solve problems and make predictions.

## 3. How do you find a linear transformation?

To find a linear transformation, you must first identify the input and output vector spaces. Then, you can represent the transformation as a matrix and use techniques such as row reduction or eigenvalue decomposition to find the specific values for the matrix elements.

## 4. What are some applications of linear transformations?

Linear transformations have many applications, including image and signal processing, machine learning, and computer graphics. They are also used in solving systems of linear equations and studying the behavior of linear systems.

## 5. Is every transformation linear?

No, not every transformation is linear. A transformation must meet certain criteria, such as preserving addition and scalar multiplication, to be considered linear. Non-linear transformations can involve functions such as exponentials, logarithms, and trigonometric functions.

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