Question

If \(A\) is an \(m \times n\) matrix, let \(C_{k}(A)\) denote column \(k\) of \(A .\) Show that \(C_{k}: \mathbf{M}_{m n} \rightarrow \mathbb{R}^{m}\) is a linear transformation for each \(k=1, \ldots, n\)

Short answer

\(C_k: \mathbf{M}_{m\times n} \rightarrow \mathbb{R}^m\) is a linear transformation because it satisfies linearity properties.

Step-by-step solution

Understand the Problem

We are given a matrix function that takes a column from an \(m \times n\) matrix \(A\) and outputs an \(m\)-dimensional vector. Our goal is to show that this function is a linear transformation.

Define Linear Transformation

A function \(T: V \to W\) is a linear transformation if for all vectors \(u, v \in V\) and scalars \(c, d \in \mathbb{R}\), it holds that \(T(cu + dv) = cT(u) + dT(v)\).

Key concepts

Matrix Functions

A matrix function, like the one given in the exercise, maps one set of matrices to another. In this case, our function maps an entire matrix to one of its columns.
Imagine you have a matrix full of numbers, and each number sits in its own little box; the columns and rows create a big grid of these boxes. When we say "matrix functions," we're talking about operations that can do things like change the size of the matrix, rearrange the elements, or focus on a specific part, such as a column, and pick it out.

In the exercise, the matrix function we're focusing on, denoted by \(C_k(A)\), picks out the \(k\)-th column of the matrix \(A\). This is a simple but very useful operation in linear algebra because columns can represent vectors. Therefore, taking out a column can help us do calculations with the vectors directly.

• Column functions are straightforward: they take a vertical slice (a column) of the matrix.
• These functions transform a 2D matrix into a 1D vector.

This kind of extraction is foundational when dealing with more complex matrix operations and is a building block for understanding how linear transformations work in linear algebra.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, and linear transformations. It's a powerful tool used in various fields like computer science, physics, and engineering. One of its fundamental concepts is the study of how we can represent and manipulate linear systems.

In our exercise, we're particularly dealing with linear transformations. These transformations map vectors from one space to another in a linear fashion. This simply means that when you apply the transformation to a combination of vectors and scale them, you can predict the outcome exactly based on the same combination and scale of their images.

• A function is a linear transformation if it preserves scalar multiplication and addition.
• This means that the structure stays consistent through transformation.

The function \(C_k\) in our problem illustrates this concept perfectly. It takes a whole matrix and outputs a single column vector, maintaining the properties of linearity. This is what makes linear algebra such a vital area of study: understanding these transformations helps us solve both simple systems and complex real-world problems.

Vector Spaces

Vector spaces are the setting for most of the work done in linear algebra. They are collections of vectors that are closed under vector addition and scalar multiplication. Think of vector spaces as playgrounds where vectors can act and interact without leaving the space.

In our exercise, the result of the matrix function \(C_k\) is a vector from a vector space \(\mathbb{R}^m\). This means the output of this function behaves as part of a set that's closed for the operations defined in vector spaces.

• Each vector space has a defined dimension, which tells us how many vectors are needed to form the space.
• Vectors in this space can be manipulated under the rules of linear transformations.

Understanding vector spaces helps you know the extent and limitations of transformations. It's crucial because they tell us how transformations like \(C_k\) maintain the integrity of data when operating within the rules of linear algebra. Vector spaces allow for the elegant and efficient handling of problems, making the combinations and resolutions systematic and predictable.