Finding Coordinate Matrix Of A Vector Relative To Standard Basis

Jul 13, 2025 by ADMIN 65 views

$Finding the Coordinate Matrix of X Relative to the Standard Basis for M_{3,1}$

In linear algebra, coordinate matrices provide a powerful way to represent vectors in different bases. Understanding how to find the coordinate matrix of a vector relative to a specific basis is crucial for various applications, including solving systems of linear equations, performing linear transformations, and analyzing vector spaces. In this comprehensive guide, we will delve into the process of determining the coordinate matrix of a vector X relative to the standard basis for M_{3,1}, the vector space of 3x1 matrices (column vectors). This article will break down the concepts, provide step-by-step instructions, and offer illustrative examples to ensure a clear understanding of the topic. Whether you are a student learning linear algebra or a professional seeking a refresher, this guide will equip you with the knowledge and skills to confidently tackle coordinate matrix problems.

Before diving into the specifics of finding the coordinate matrix, it's essential to establish a solid foundation of the underlying concepts. Let's begin by defining some key terms and principles.

What is a Vector Space?

A vector space is a collection of objects, called vectors, that can be added together and multiplied by scalars (usually real numbers) while still remaining within the same space. These operations must satisfy certain axioms, ensuring the space behaves predictably. Common examples of vector spaces include the set of all n-tuples of real numbers (R^n), the set of all m x n matrices (M_{m,n}), and the set of all polynomials of degree n or less.

What is a Basis?

A basis for a vector space is a set of linearly independent vectors that span the entire space. This means that any vector in the space can be written as a unique linear combination of the basis vectors. A basis provides a fundamental framework for representing all vectors within a vector space efficiently.

Standard Basis for M_{3,1}

The standard basis for M_{3,1}, the vector space of 3x1 matrices (column vectors), is a specific set of three linearly independent vectors that span the space. These vectors are:

e_1 = 
\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, 
e_2 = 
\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}, 
e_3 = 
\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}

These vectors are called the standard basis vectors because they have a 1 in one position and 0s in all other positions. They form the most natural and straightforward basis for M_{3,1}.

Coordinate Matrix

The coordinate matrix of a vector X relative to a basis B is a column vector that lists the coefficients needed to express X as a linear combination of the basis vectors in B. In other words, it tells us how much of each basis vector we need to add together to obtain the vector X.

Given a vector X and a basis B = {b_1, b_2, ..., b_n}, the coordinate matrix of X relative to B, denoted as [X]_B, is the column vector:

[X]_B = 
\begin{bmatrix}
c_1 \\
c_2 \\
... \\
c_n
\end{bmatrix}

where X = c_1b_1 + c_2b_2 + ... + c_n*b_n.

Now that we have a grasp of the fundamental concepts, let's revisit the problem at hand. We are given a vector:

X = 
\begin{bmatrix}
0 \\
2 \\
1
\end{bmatrix}

and we are tasked with finding the coordinate matrix of X relative to the standard basis for M_{3,1}.

To find the coordinate matrix of X relative to the standard basis, we need to express X as a linear combination of the standard basis vectors and identify the coefficients.

Step 1: Express X as a Linear Combination of the Standard Basis Vectors

We want to find scalars c_1, c_2, and c_3 such that:

X = c_1*e_1 + c_2*e_2 + c_3*e_3

Substituting the given vector X and the standard basis vectors, we have:

\begin{bmatrix}
0 \\
2 \\
1
\end{bmatrix} = c_1 * \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix} + c_2 * \begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix} + c_3 * \begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}

Step 2: Determine the Coefficients

Performing the scalar multiplication and vector addition on the right-hand side, we get:

\begin{bmatrix}
0 \\
2 \\
1
\end{bmatrix} = \begin{bmatrix}
c_1 \\
0 \\
0
\end{bmatrix} + \begin{bmatrix}
0 \\
c_2 \\
0
\end{bmatrix} + \begin{bmatrix}
0 \\
0 \\
c_3
\end{bmatrix} = \begin{bmatrix}
c_1 \\
c_2 \\
c_3
\end{bmatrix}

By equating the corresponding entries of the matrices, we can directly read off the coefficients:

c_1 = 0
c_2 = 2
c_3 = 1

Step 3: Construct the Coordinate Matrix

Now that we have the coefficients, we can construct the coordinate matrix of X relative to the standard basis. The coordinate matrix is simply a column vector formed by the coefficients:

[X]_B = 
\begin{bmatrix}
c_1 \\
c_2 \\
c_3
\end{bmatrix} = 
\begin{bmatrix}
0 \\
2 \\
1
\end{bmatrix}

Therefore, the coordinate matrix of X relative to the standard basis for M_{3,1} is:

[X]_B = 
\begin{bmatrix}
0 \\
2 \\
1
\end{bmatrix}

In this guide, we have thoroughly explored the process of finding the coordinate matrix of a vector X relative to the standard basis for M_{3,1}. We began by establishing a solid understanding of key concepts such as vector spaces, bases, and coordinate matrices. We then walked through a step-by-step solution, illustrating how to express X as a linear combination of the standard basis vectors and extract the coefficients to form the coordinate matrix. Mastering this process is essential for anyone working with linear algebra, as it provides a fundamental tool for representing vectors in different bases and performing various linear transformations. By understanding the principles and following the outlined steps, you can confidently tackle similar problems and deepen your understanding of linear algebra.

To further solidify your understanding, try solving the following practice problems:

Find the coordinate matrix of Y = \begin{bmatrix} -1 \ 3 \ 2 \end{bmatrix} relative to the standard basis for M_{3,1}.
Find the coordinate matrix of Z = \begin{bmatrix} 4 \ -2 \ 0 \end{bmatrix} relative to the standard basis for M_{3,1}.

If you're eager to delve deeper into linear algebra, consider exploring the following topics:

Change of basis
Linear transformations
Eigenvalues and eigenvectors
Inner product spaces

By continuing your exploration, you'll unlock even more powerful tools and techniques for solving complex problems in mathematics, engineering, and computer science.

Coordinate matrix, standard basis, vector space, linear combination, basis vectors, M_{3,1}, linear algebra, matrix representation, vector representation, coefficients.