Question

If $T:V\to V$ is a linear transformation, find $T(\mathbf{v})$ and $T(\mathbf{w})$ if: a. $T(\mathbf{v}+\mathbf{w})=\mathbf{v}-2\mathbf{w}$ and $T(2\mathbf{v}-\mathbf{w})=2\mathbf{v}$ b. $T(\mathbf{v}+2\mathbf{w})=3\mathbf{v}-\mathbf{w}$ and $T(\mathbf{v}-\mathbf{w})=2\mathbf{v}-4\mathbf{w}$

Short answer

Solution steps need to be expanded for a full answer.

Step-by-step solution

Define Equations from Given Conditions

For part a, you have two conditions: 1. $T(\mathbf{v}+\mathbf{w})=\mathbf{v}-2\mathbf{w}$ 2. $T(2\mathbf{v}-\mathbf{w})=2\mathbf{v}$. For part b, the conditions are:1. $T(\mathbf{v}+2\mathbf{w})=3\mathbf{v}-\mathbf{w}$2. $T(\mathbf{v}-\mathbf{w})=2\mathbf{v}-4\mathbf{w}$.

Key concepts

Vector Spaces

Vector spaces are mathematical structures formed by vectors. These vectors can be added together and multiplied by scalars to produce new vectors.
This concept is foundational in linear algebra, providing the framework to understand linear transformations.
A vector space must satisfy certain axioms, like:

• Associativity of vector addition: For any vectors $\mathbf{u},\mathbf{v},\mathbf{w}$, the equation $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$ holds.
• Existence of zero vector: There exists a vector $\mathbf{0}$ such that $\mathbf{v}+\mathbf{0}=\mathbf{v}$ for any vector $\mathbf{v}$.
• Existence of additive inverses: For every vector $\mathbf{v}$, there exists a vector $-\mathbf{v}$ such that $\mathbf{v}+(-\mathbf{v})=\mathbf{0}$.

Understanding these properties helps students grasp how transformations like $T$ affect vectors $(\mathbf{v},\mathbf{w})$ in a vector space.
In the original exercise, we explore how linear transformations modify vectors within the same space, maintaining their core properties.

Matrix Representations

Matrix representation is a convenient method to describe linear transformations in a structured form.
When you deal with transformations, you can often represent them using matrices.
This allows for easier computation and visualization. A linear transformation $T:V\to V$ can be expressed as a matrix $A$ if a basis for $V$ is chosen.

• The matrix $A$ acts on a vector $\mathbf{x}$ by the matrix-vector product $A\mathbf{x}$.
• For the transformation defined by conditions a and b, finding $T(\mathbf{v})$ and $T(\mathbf{w})$ can involve expressing these transformations in terms of a matrix.

In the given example, while the conditions provide information on the transformations involving combinations of $\mathbf{v}$ and $\mathbf{w}$, a matrix could potentially simplify the representation.
Matrices allow transforming complex problems into a series of simpler calculations that match the matrix's structure, fostering deeper understanding and application.

Algebraic Structures

Algebraic structures like groups, rings, and fields underpin much of linear algebra's application and theory. These structures help explain how operations are defined and combined.
A linear transformation itself can be seen through the lens of these structures as they possess properties from these fundamental constructs.

• Groups provide a basic set with operation rules; add and multiply vectors, and define inverses.
• Fields introduce scalars into the equation, essential for vector space operations.

The exercise defines $T$ through combined algebraic operations on $\mathbf{v}$ and $\mathbf{w}$.
The expressions like $\mathbf{v}+2\mathbf{w}$ show how these operations work under linear combinations. Understanding algebraic structures clarifies how transformations interact and maintain these systematic properties across defined expressions.
By grounding these transformations in algebraic principles, students can predict behaviors across more complex operations in linear algebra.