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For each linear operator \(\mathrm{T}\), find a Jordan canonical form \(J\) of \(\mathrm{T}\) and a Jordan canonical basis \(\beta\) for \(\mathrm{T}\). (a) \(V\) is the real vector space of functions spanned by the set of realvalued functions \(\left\\{e^{t}, t e^{t}, t^{2} e^{t}, e^{2 t}\right\\}\), and \(\mathrm{T}\) is the linear operator on \(\mathbf{V}\) defined by \(\mathbf{T}(f)=f^{\prime}\). (b) \(\mathrm{T}\) is the linear operator on \(\mathrm{P}_{3}(R)\) defined by \(\mathrm{T}(f(x))=x f^{\prime \prime}(x)\). (c) \(\mathrm{T}\) is the linear operator on \(\mathrm{P}_{3}(R)\) defined by \(\mathrm{T}(f(x))=f^{\prime \prime}(x)+2 f(x)\). (d) \(\mathrm{T}\) is the linear operator on \(\mathrm{M}_{2 \times 2}(R)\) defined by $$ \mathrm{T}(A)=\left(\begin{array}{ll} 3 & 1 \\ 0 & 3 \end{array}\right) \cdot A-A^{t} . $$ (e) \(\mathrm{T}\) is the linear operator on \(\mathrm{M}_{2 \times 2}(R)\) defined by $$ \mathrm{T}(A)=\left(\begin{array}{ll} 3 & 1 \\ 0 & 3 \end{array}\right) \cdot\left(A-A^{t}\right) . $$ (f) \(V\) is the vector space of polynomial functions in two real variables \(x\) and \(y\) of degree at most 2, as defined in Example 4, and T is the linear operator on \(V\) defined by $$ \mathrm{T}(f(x, y))=\frac{\partial}{\partial x} f(x, y)+\frac{\partial}{\partial y} f(x, y) . $$

Step by step solution

01

Matrix of T with respect to the basis

Let's consider the basis \(\left\\{e^{t}, t e^{t}, t^{2} e^{t}, e^{2 t}\right\\}\). We compute the action of T on each function in the basis and write its matrix: $$ T(e^t) = e^t \\ T(te^t) = e^t + te^t\\ T(t^2e^t) = 2te^t + t^2e^t\\ T(e^{2t}) = 2e^{2t} $$ The matrix representation of T with respect to the basis is: $$ A = \(\begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{pmatrix}\)

02

Characteristic and minimal polynomials

The characteristic polynomial of A is: \(\det (A - \lambda I) = (\lambda - 1)^3(\lambda - 2)\). The minimal polynomial is: \(m_A(\lambda) = (\lambda - 1)^2(\lambda - 2)\).

03

Eigenvalues and their algebraic multiplicities

From the characteristic polynomial, there are two eigenvalues: 位鈧� = 1 with algebraic multiplicity 3 and 位鈧� = 2 with algebraic multiplicity 1.

04

Dimensions of eigenspaces

The eigenspace for 位鈧� = 1 is given by: \(E_{1} = \left\{\boldsymbol{v} \in V : (A - I)\boldsymbol{v} = \boldsymbol{0}\right\}\), and it is one-dimensional. The eigenspace for 位鈧� = 2 is given by: \(E_{2} = \left\{\boldsymbol{v} \in V : (A - 2I)\boldsymbol{v} = \boldsymbol{0}\right\}\), and it is also one-dimensional.

05

Jordan canonical basis and the Jordan form

We can now find the Jordan canonical basis. We take one eigenvector from the eigenspace of 位鈧� = 1 (say, \(\boldsymbol{v}_1\)) and build the vector \(\boldsymbol{v}_2 = A\boldsymbol{v}_1 - \boldsymbol{v}_1\). We take one eigenvector from the eigenspace of 位鈧� = 2 (say, \(\boldsymbol{v}_3\)) and build the vector \(\boldsymbol{v}_4 = A\boldsymbol{v}_3 - 2\boldsymbol{v}_3\). The Jordan canonical basis is given by: \(\boldsymbol{v}_1 = \(\begin{pmatrix}1 \\ 0 \\ 0 \\ 0\(\end{pmatrix}, \boldsymbol{v}_2 = \(\begin{pmatrix}1\\ 1\\ 0\\ 0\), \boldsymbol{v}_3 = \(\begin{pmatrix}0\\ 0\\ 0\\ 1\(\end{pmatrix}, \boldsymbol{v}_4 = \(\begin{pmatrix}0 \\ 0 \\ 0 \\ 2 \(\end{pmatrix}\)\). The Jordan form J is given by: \(J = \(\begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 2 \(\end{pmatrix}\)\`. The solutions for the other operators can be similarly computed by following the same steps, being sure to adjust the matrix computations to suit each specific linear operator.

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Key Concepts

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

Linear Operator

A linear operator is a type of function that maps elements from one vector space to another while preserving the operations of vector addition and scalar multiplication. To visualize a linear operator, think of it as a machine where you input a vector from one space and receive a transformed vector in another space, or possibly the same space.

Linear operators are foundational in understanding complex mathematical concepts because they provide a systematic way to transform and manipulate vectors. For instance, in the exercise provided, the operator \(\mathrm{T}\) is defined in various ways across different vector spaces, such as real functions or polynomials. Each definition specifies a rule for transforming an input function or matrix.

Eigenvalues

Imagine having a particular type of vector that, when a linear operator acts on it, merely stretches or shrinks it without changing its direction. These special vectors are related to eigenvalues, which are scalars indicating the magnitude of that stretching or shrinking. To formally find the eigenvalues of a linear operator, we calculate the roots of the characteristic polynomial.

In the context of the exercise, eigenvalues are crucial to understand because they help us determine the behavior of the linear operator and how it might simplify under certain conditions. For instance, when deriving the Jordan canonical form, the eigenvalues guide us on how to construct the Jordan blocks.

Eigenspaces

Closely linked to eigenvalues are eigenspaces. Each eigenvalue has an associated eigenspace, which is the set of all vectors that, after being transformed by the linear operator, remain on the same line as the initial vector. In other words, the eigenspace is a subspace consisting of all the eigenvectors corresponding to a particular eigenvalue, along with the zero vector.

In our exercise, the determination of eigenspaces involves solving homogeneous equations like \( (A - \lambda I)\boldsymbol{v} = \boldsymbol{0} \) for each eigenvalue \(\lambda\). The number of linearly independent solutions to these equations tells us the dimension of the eigenspaces, which is a key step in constructing the Jordan canonical basis.

Characteristic Polynomial

The characteristic polynomial is a polynomial that plays a pivotal role in grasping the properties of a linear operator. It is obtained by taking the determinant of the matrix representation of the linear operator subtracted by a scalar \(\lambda\) multiplied by the identity matrix, denoted as \(A - \lambda I\).

The roots of this polynomial give us the eigenvalues of the operator. In the exercise, once the characteristic polynomial was determined for the matrix \(A\), it became possible to identify the eigenvalues and subsequently the structure of the Jordan canonical form. Understanding how to compute and interpret the characteristic polynomial is essential for analyzing the behavior of linear operators.