Discussion-4

Exercise LT.C43

Consider the linear transformation \[ T:p_3 \rightarrow p_2\\by\\ T(ax+bx+cx^{2}+dx^{3})=b+2cx+3dx^{2} \] First, let’s find the pre-image of 0 under this transformation. Does this linear transformation seem familiar?

Lets understand what the pre-image is http://mathworld.wolfram.com/Pre-Image.html

A Pre-image exists if whether f has an inverse or not. If f does not have an inverse, then the preimage is defined as the exact image of y.

Let the following be the preimage of T \[ T^{-1}(0) \]

The preimage of T is the set of all polynomials where the following holds true \[ T(ax+bx+cx^{2}+dx^{3})=0 \]

This implies that the following is also true \[ b+2cx+3dx^{2}=0 \] Therefore it can be said that zero represents the 0 polynomial, in otherwords the set of all polynomials where a=0,b=0,and c=0 thus polynomials of degree 0.

How does this transformation look familiar? \[ T(ax+bx+cx^{2}+dx^{3})=b+2cx+3dx^{2} \] Is the same as \[ dx/dy(ax+bx+cx^{2}+dx^{3})=b+2cx+3dx^{2} \]

In other words, the expression on the right is the derivative of the polynomial on the left assuming a, b, and c are non zero constants. The derivative of a constant is zero. The derivative of the other terms can be found using the power rule. \[ dx/dy(ax^{n})=nax^{n-1} \]

# Define the linear transformation function T
T <- function(poly) {
  return(poly[2] + 2 * poly[3] * seq_along(poly)[-1] + 3 * poly[4] * (seq_along(poly)[-1])^2)
}

# Find the pre-image of 0
preimage <- function() {
  # Coefficients of the polynomial
  a <- 0
  b <- 0
  c <- 0
  d <- 0
  
  # Check if T(ax+bx+cx^2+dx^3) = 0
  result <- b + 2 * c * seq(0, 2) + 3 * d * (seq(0, 2))^2
  
  # If all coefficients are 0, it's the zero polynomial
  if(all(result == 0)) {
    return("The pre-image of 0 is the zero polynomial")
  } else {
    return("There is no pre-image of 0")
  }
}

# Function to show the familiar nature of the transformation
familiarity <- function() {
  # Coefficients of the polynomial
  a <- 1
  b <- 2
  c <- 3
  d <- 4
  
  # Derivative of the polynomial
  derivative <- c(0, a, 2 * b, 3 * c)  # Derivative of ax + bx + cx^2 + dx^3
  
  # Transformation using T
  transformed <- T(c(0, a, b, c, d))
  
  # Check if the derivative and the transformed polynomial match
  if(all(derivative == transformed)) {
    return("The transformation resembles taking the derivative of the polynomial with respect to y")
  } else {
    return("The transformation does not resemble taking the derivative of the polynomial with respect to y")
  }
}

# Print the results
print(preimage())

## [1] "The pre-image of 0 is the zero polynomial"

print(familiarity())

## [1] "The transformation does not resemble taking the derivative of the polynomial with respect to y"