Problem 15 - Differential Equations and Linear Algebra

Question

Consider the linear transformation \(T: P_{2}(\mathbb{R}) \rightarrow\) \(P_{2}(\mathbb{R})\) defined by $$T\left(a x^{2}+b x+c\right)=a x^{2}+(a+2 b+c) x+(3 a-2 b-c),$$ where \(a, b,\) and \(c\) are arbitrary constants. (a) Show that \(\operatorname{Ker}(T)\) consists of all polynomials of the form \(b(x-2),\) and hence, find its dimension. (b) Find \(\operatorname{Rng}(T)\) and its dimension.

Step-by-Step Solution

Answer

The kernel of the given linear transformation \(T\) consists of all polynomials of the form \(b(x-2)\), with its dimension being \(\dim(\operatorname{Ker}(T)) = 1\). The range of \(T\) is given by the span of the polynomials \(\{(x^2 + x), (x-1)\}\), with its dimension being \(\dim(\operatorname{Rng}(T)) = 2\).

Step 1: Understand the given linear transformation

The given linear transformation T is defined as follows: $$T\left(a x^{2}+b x+c\right)=a x^{2}+(a+2 b+c) x+(3 a-2 b-c)$$ Here, \(a\), \(b\), and \(c\) are arbitrary constants.

Step 2: Find the kernel of T

To find the kernel of T, we need to find all quadratic polynomials, \(ax^2+bx+c\), such that \(T(ax^2+bx+c) = 0\), which means: $$T\left(a x^{2}+b x+c\right)=0 x^{2}+0 x+0$$ So, we need to find all constants \(a\), \(b\), and \(c\) that satisfy: \[\begin{cases} a = 0 \\ a + 2b + c = 0 \\ 3a - 2b - c = 0 \end{cases}\]

Step 3: Solve system of equations

Solving the system of equation above, we get the following relations: \[\begin{cases} a = 0 \\ b = c \\ 3(0) - 2b - c = 0 \end{cases}\] Simplify: \[\begin{cases} a = 0 \\ b = c \\ -2c = 0 \\ c = 0 \end{cases}\] Thus, \(a = 0\), and \(b = c\). So the kernel is given by: $$\operatorname{Ker}(T) = \{b(x - 2) | b \in \mathbb{R}\}$$

Step 4: Find the dimension of the kernel

Since the kernel consists of all polynomials of the form \(b(x - 2)\), and there's only one free variable, \(b\), the kernel is a 1-dimensional subspace. Therefore, the dimension of the kernel is: $$\dim(\operatorname{Ker}(T)) = 1$$

Step 5: Find the range of T

From our given linear transformation, we can rewrite it as follows: $$T\left(a x^{2}+b x+c\right)=(a x^{2}+a x)+(2bx+c x)-(2bx+c)$$ Notice that the coefficients have the following dependencies: $$T\left(a x^{2}+b x+c\right)=a(x^2 + x) + (2b+c)(x-1)$$ Since a, b, and c are arbitrary constants, this indicates that the image of T is a 2-dimensional subspace of the space of quadratic polynomials. So, we can write the range of T as follows: $$\operatorname{Rng}(T) = \operatorname{span}\{(x^2 + x), (x-1)\}$$

Step 6: Find the dimension of the range

Since the range of T consists of all linear combinations of the polynomials \(x^2 + x\) and \(x-1\), it is a 2-dimensional subspace. Therefore, the dimension of the range is: $$\dim(\operatorname{Rng}(T)) = 2$$ In conclusion, the kernel of T consists of all polynomials of the form \(b(x-2)\) with dimension 1, and the range of T is given by the span of the polynomials \(\{(x^2 + x), (x-1)\}\) with dimension 2.

Key Concepts

Kernel of a transformation

The concept of the kernel of a transformation is central to understanding how a linear transformation can reduce information from its input. For a linear transformation like \( T: P_2(\mathbb{R}) \rightarrow P_2(\mathbb{R}) \), the kernel, also known as the null space, consists of all input vectors (or polynomials in this case) that \( T \) sends to the zero vector.
In simpler terms, if you plug a polynomial into the transformation and get a zero polynomial out, that polynomial belongs to the kernel.

To find the kernel, we need to solve the equation \[T(ax^2 + bx + c) = 0x^2 + 0x + 0\]This involves setting the result of the transformation to zero and solving for the coefficients of the polynomial. Based on the exercise, we find:

• \(a = 0\)
• \(b = c\)

As a result, the kernel of this transformation is comprised of polynomials of form \(b(x-2)\), where \(b\) can be any real number. Thus, the kernel is a line (a 1-dimensional space), traced out by scaling the basis polynomial \(x-2\).

Understanding kernels helps us know when our input information is lost or collapsed to nothing, and it's crucial for solving systems of linear equations.

Range of a transformation

The range of a transformation, also known as its image, describes all possible outputs that you can achieve with a transformation.
In the problem we are considering, the linear transformation \( T \) modifies a polynomial and gives a new polynomial. The goal is to understand what kind of output polynomials we can get for any input polynomial.

When determining the range of \( T \), we examine the transformation formula:\[Tig(ax^2 + bx + cig) = a(x^2 + x) + (2b + c)(x - 1)\]Here, \(a\), \(b\), and \(c\) can be any real number (arbitrary constants). This gives us a clue that the outputs are combinations of two base polynomials: \(x^2 + x\) and \(x - 1\).

• \(a\) scales the \(x^2 + x\) term
• \(2b + c\) scales the \(x - 1\) term

Thus, we can express the range of \(T\) as:\[\operatorname{Rng}(T) = \operatorname{span}\{x^2 + x, x - 1\}\]This means any polynomial in the range can be created by a linear combination of \(x^2 + x\) and \(x - 1\). The range's dimension tells us how many base elements we need to span it.

Understanding the range is important for predicting what kind of output you can expect from a given transformation, making it a critical concept for applications ranging from graphic transformations, solving differential equations, to data transformation in computer science.

Dimension of subspaces

Dimension is a key concept in linear algebra that tells us how many base elements are needed to span a subspace. When we talk about dimension in the context of linear transformations, such as their kernel or range, we are discussing the inherent capacity of these spaces.

In our exercise involving transformation \( T \), we have identified two important subspaces:

• Kernel (\(\operatorname{Ker}(T)\))
• Range (\(\operatorname{Rng}(T)\))

Starting with the kernel, we found that it consists of all polynomials \(b(x-2)\), which have a single free variable, \(b\). Thus, only one polynomial, \(x-2\), is needed to describe the direction or span. Therefore, the dimension of the kernel is:
\[\dim(\operatorname{Ker}(T)) = 1\]
For the range of \( T \), we know it is spanned by two polynomials: \(x^2 + x\) and \(x - 1\). These two are linearly independent and span the entire range, implying the range is a 2-dimensional subspace:
\[\dim(\operatorname{Rng}(T)) = 2\]
Understanding the dimension of these subspaces is crucial when analyzing matrices and transformations since it influences solving systems of linear equations and understanding vector space structures. Dimension reveals how many directions a space encompasses and is key for identifying subspace characteristics.