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Chapter 10: Problem 20

Let \(T: \mathbb{R}_{3}[x] \rightarrow M_{2}(\mathbb{R})\) be given by $$ T\left(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\right)=\left[\begin{array}{ll} a_{0}+a_{1} & a_{1}+a_{2} \\ a_{2}+a_{3} & a_{3}+a_{0} \end{array}\right] $$ Check that \(T\) is linear. What is its kernel? Determine a basis for its image.

Short Answer

Expert verified
T is linear. The kernel is spanned by \(\{ 1 - x + x^2 - x^3 \}\). The image basis is \(\{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix} \}\).

Step by step solution

01

Understanding Linearity

A transformation \( T: V \to W \) is linear if for all \( u, v \in V \) and scalars \( c \in \mathbb{R} \), \( T(u + v) = T(u) + T(v) \) and \( T(cu) = cT(u) \). Let's verify these properties for the given transformation.
02

Checking Additivity

Let \( p(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 \) and \( q(x) = b_0 + b_1 x + b_2 x^2 + b_3 x^3 \) be two polynomials in \( \mathbb{R}_3[x] \). Then,\[ T(p(x) + q(x)) = T((a_0+b_0) + (a_1+b_1)x + (a_2+b_2)x^2 + (a_3+b_3)x^3) \]By transforming,\[ \begin{bmatrix} (a_0+b_0) + (a_1+b_1) & (a_1+b_1) + (a_2+b_2) \ (a_2+b_2) + (a_3+b_3) & (a_3+b_3) + (a_0+b_0) \end{bmatrix} \]This equals \( T(p(x)) + T(q(x)) \), confirming additivity.
03

Checking Homogeneity

Now check for a scalar \( c \) and polynomial \( p(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 \).\[ T(c \cdot p(x)) = T(ca_0 + ca_1 x + ca_2 x^2 + ca_3 x^3) \]\[ = \begin{bmatrix} ca_0 + ca_1 & ca_1 + ca_2 \ ca_2 + ca_3 & ca_3 + ca_0 \end{bmatrix} = c \cdot T(p(x)) \]This confirms homogeneity, thus \( T \) is linear.
04

Finding the Kernel

The kernel of \( T \) is the set of all polynomials \( p(x) \) such that \( T(p(x)) = 0 \). Set,\[ \begin{bmatrix} a_0 + a_1 & a_1 + a_2 \ a_2 + a_3 & a_3 + a_0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \]From the equations, 1. \( a_0 + a_1 = 0 \)2. \( a_1 + a_2 = 0 \)3. \( a_2 + a_3 = 0 \)4. \( a_3 + a_0 = 0 \)Solving these gives \( a_0 = -a_1, a_1 = -a_2, a_2 = -a_3 \), so the solution is \( \text{span} \{ 1 - x + x^2 - x^3 \} \).
05

Determining the Image and its Basis

The image of \( T \) consists of all matrices that can be formed by varying the coefficients \( a_0, a_1, a_2, a_3 \) in \[ \begin{bmatrix} a_0 + a_1 & a_1 + a_2 \ a_2 + a_3 & a_3 + a_0 \end{bmatrix} \].Considering each coefficient inline with the unit polynomial vectors:- \( a_0 = 1, a_1 = 0, a_2 = 0, a_3 = 0 \): \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \)- \( a_0 = 0, a_1 = 1, a_2 = 0, a_3 = 0 \): \( \begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix} \)- \( a_0 = 0, a_1 = 0, a_2 = 1, a_3 = 0 \): \( \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \)- \( a_0 = 0, a_1 = 0, a_2 = 0, a_3 = 1 \): \( \begin{bmatrix} 0 & 0 \ 1 & 1 \end{bmatrix} \)These matrices form a basis for the image.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Algebra
Linear algebra is a fundamental branch of mathematics that deals with vectors, vector spaces, and linear transformations. It provides the tools for modeling and solving problems in engineering, physics, computer science, and other fields. A key concept is the idea of a linear transformation, which maps vectors from one vector space to another while preserving vector addition and scalar multiplication. This means that linear transformations respect the structure of the space, making them predictable and useful.
  • Vector Space: A set of vectors where addition and scalar multiplication are defined and follow certain rules.
  • Linear Transformation: A function between two vector spaces that preserves addition and scalar multiplication.
  • Matrix Representation: Matrices can represent linear transformations, making computations manageable.
These concepts form the backbone of many modern mathematical applications, such as 3D graphics, data analysis, and machine learning. By understanding linear algebra, you gain a powerful toolkit for approaching complex problems.
Kernel of a Transformation
The kernel of a transformation, especially in linear algebra, is a set of input vectors that are mapped to the zero vector in the output space. It is a crucial concept because it provides insight into the properties and behavior of a transformation.To find the kernel of the transformation \( T \), you solve for all vectors \( p(x) \) that satisfy \( T(p(x)) = 0 \). In this problem, the transformation is a mapping from the polynomial space \( \mathbb{R}_3[x] \) to the space of 2x2 matrices. The kernel is the solution set of polynomials that result in the zero matrix:
  • Identify equations based on zero matrix conditions.
  • Set each entry of the resulting matrix to zero.
  • Solve the system of equations to find the specific form of polynomials that map to zero.
Understanding the kernel helps you grasp the constraints and limitations of the transformation, showing how certain parts of the input space are effectively 'lost' when transformed into the output space.
Basis of Image
The image of a transformation is the set of all output vectors that can be obtained by applying the transformation to every possible input vector. Finding a basis for the image is like finding the minimum set of vectors that can generate all outputs of the transformation. This set provides a compact representation of the transformation's range.To determine a basis for the image of \( T \), you:
  • Vary the coefficients of the input polynomial \( a_0, a_1, a_2, a_3 \) systematically.
  • Observe the resulting matrices to see different possible outputs.
  • Select independent output matrices that span all possible outputs.
By identifying a basis, we better understand how the transformation affects different inputs and can predict the form of any output, which is essential for applications in solving linear systems or simulating systems in physics.
Additivity and Homogeneity
Additivity and homogeneity are two fundamental properties that define a linear transformation. Ensuring these properties hold is essential to confirming the transformation's linearity.Additivity ensures that the transformation of the sum of two vectors is the same as the sum of their transformations:
  • For polynomials \( p(x) \) and \( q(x) \), \, \( T(p(x) + q(x)) = T(p(x)) + T(q(x)) \).
  • Verify this by transforming both sides and confirming equality.
Homogeneity concerns scaling of vectors before and after transformation:
  • For a scalar \( c \) and polynomial \( p(x) \), \, \( T(c \cdot p(x)) = c \cdot T(p(x)) \).
  • Check by applying the transformation to \( c \cdot p(x) \) and ensuring it equals \( c \) times \( T(p(x)) \).
These properties are crucial as they maintain the structure and properties of vector spaces during transformations, making computations consistent and reliable across applications.

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Most popular questions from this chapter

(a) Let \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be defined by \(T(x, y)=(2 x+3 y+4,5 x-y)\). Show that \(T\) is not linear. (b) Let \(p(x)\) be a fixed polynomial in \(\mathbb{R}[x]\). Define \(T: \mathbb{R}[x] \rightarrow \mathbb{R}[x]\) by: for each polynomial \(q(x) \in \mathbb{R}[x], T(q(x))=p(x) q(x)\). Is \(T\) a linear map?

Is there an isomorphism \(T: M_{2}(\mathbb{R}) \rightarrow \mathbb{R}_{3}[x]\) in which $$ T\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]=1+x+x^{3} ? $$ Either prove there is not or define \(T\) completely by writing down $$ T\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right], \quad T\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right] \text { and } T\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right] $$

Let \(T: \mathbb{R}^{5} \rightarrow \mathbb{R}^{4}\) be the map defined by: $$ T(x, y, z, w, t)=(x+y+2 z, x+2 y+4 z+2 w+t, 2 z+2 w+t, 2 x+2 y-4 w-2 t) $$ Do you believe that \(T\) is linear? Give the reasons for your belief. Now, verify your belief by means of a proof or a counterexample.

Let \(C\) be a fixed invertible matrix from \(M_{2}(\mathbb{R})\). Define the map \(T: M_{2}(\mathbb{R}) \rightarrow M_{2}(\mathbb{R})\) by \(T(A)=C^{-1} A C\) for all \(A \in M_{2}(\mathbb{R})\). Is \(T\) linear?

Is there a linear transformation \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{4}\) such that \(T(1,2,3)=(0,1,0,1)\), \(T(2,3,4)=(2,-3,1,7)\) and \(T(3,4,5)=(4,-7,2,14)\) ? If not, why not?
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