Algebra lineal - Tema IV Linear Transformations-1

Introduction to Linear Transformations

Topic IV: Linear Transformations

This section covers the essential concepts related to linear transformations in the context of Linear Algebra, as taught at laSalle Universitat Ramon Llull.

Definition and Properties of Linear Transformations

A transformation is defined as a relation between two sets where each element of the original set maps to exactly one element of the destination set, known as the image. A linear transformation is defined for vector spaces E and F over a commutative field K. A transformation ( f: E \to F ) is considered linear if it satisfies two conditions: additivity, expressed as ( f(u+v) = f(u) + f(v) ) for all ( u, v ) in E, and homogeneity, which states ( f(\alpha u) = \alpha f(u) ) for all scalars ( \alpha ) in K and vectors ( u ) in E.

Examples of Linear Transformations

For instance, consider the example of a linear transformation ( f: \mathbb{R}^2 \to \mathbb{R} ), defined as ( f(u) = x + y ) for ( u = (x, y) ). This function satisfies the conditions of additivity and homogeneity. Conversely, the transformation defined by ( f(u) = xy ) does not satisfy the properties of a linear transformation.

Properties of Linear Transformations

Several important properties of linear transformations can be established. Property 1 addresses the generalization of linear transformation results. Property 2 notes that the image of the zero vector in E is the zero vector in F, proved by showing ( f(0) = 0_F ) for all vectors ( u_1, u_2 ) in E. Additionally, Property 3 states symmetry: ( f(-u) = -f(u) ). Property 4 confirms that the composition of two linear transformations is itself linear.

Further properties include Property 5, which states that the image of a subspace is a subspace of F, and Property 6 asserts that the preimage of a subspace in F, provided that subspace is within the image of E, is a subspace of E.

Kernel of a Linear Transformation

The kernel, denoted as Ker f, is defined as the set of all vectors in E that map to the zero vector in F. Formally, Ker f can be represented as ( { u \in E | f(u) = 0_F } ). Notably, the kernel is a subspace of E.

Examples of Kernels

In illustrating kernels, one example includes a linear transformation defined by ( f(u) = (x + z, x + y + z) ) from ( \mathbb{R}^3 ) to ( \mathbb{R}^2 ), where the basis and dimension of the kernel must be determined. Another example involves transformations using matrix operations that require finding bases and dimensions of the kernel.

Image of a Linear Transformation

The image, denoted as Im f, represents the set of all images of vectors from E and is expressed as ( \text{Im} f = { v \in F | \exists u \in E \text{ such that } f(u) = v } ).

Finding Images

Exercises provided focus on computing bases and dimensions of the image based on specific transformations involving vectors and operations.

Propositions and Definitions

Proposition 1 relates to the dimensions of the kernel and image, positing that the dimension of E equals the sum of the dimensions of the kernel and image, i.e., ( \text{dim} E = \text{dim} \text{Ker} f + \text{dim} \text{Im} f ). This relationship is essential for evaluating linear transformations.

Finding Bases and Dimensions

A methodological approach is utilized to determine the basis of the image based on given dimensions and relationships of kernels, which often involves checking the rank of transformations.

Composite Linear Transformations

Composite linear transformations refer to composing two transformations to yield another linear transformation, illustrating how each transformation affects the other.

Matrix Representation of Transformations

The text explains how linear transformations are associated with matrices, including their representations and how composition can be represented using matrix multiplication.

Properties of Associated Matrices

The properties of associated matrices indicate that the columns represent images of the input basis expressed in the output basis, highlighting the relationships between dimensions.

Associating Kernels and Images with Matrices

Exercises emphasize computing associated matrices to linear transformations, aimed at understanding the implications on kernels and images.

The Rouché-Frobenius Theorem

This theorem connects linear transformations with systems of linear equations and the structure of their solutions, which is vital for grasping the roles of kernels and images in practical applications.

Inverses of Linear Transformations

The discussion on inverses addresses bijective transformations, injectivity, and surjectivity while considering their implications on the dimensions of kernel and image.

Exercises on Linear Maps

Practical assignments are provided to emphasize how linear maps can be verified against specific scenarios and conditions.

Final Notes on Linear Transformations

The concluding remarks emphasize the relationship between linear algebraic concepts and their geometric interpretations, reinforcing the foundational importance of linear transformations in advanced mathematics.