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Chapter 4: Problem 61

Determine the dimension of the vector space. $$M_{23}$$

Short Answer

Expert verified
The dimension of the matrix space \(M_{23}\) is 6.

Step by step solution

01

Understand the matrix properties

The given matrix is \(M_{23}\), which means it has 2 rows and 3 columns. This is a matrix that resides in a space of two-dimensional matrices.
02

Calculate the dimension

The dimension of matrix space \(M_{23}\) is obtained by multiplying the number of rows and columns. So, the dimension of the given matrix is 2*3.

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Key Concepts

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

Matrix Properties
Understanding matrix properties is crucial when dealing with vector spaces in mathematics.
  • A matrix is simply an arrangement of numbers (or sometimes other mathematical objects) in rows and columns.
  • Each matrix has dimensions, which tell you how many rows and columns it contains.
  • For instance, a matrix labeled as \(M_{23}\) indicates that it has 2 rows and 3 columns, forming what's known as a "two-dimensional matrix."
  • These dimensions play a vital role in defining the properties and behavior of the matrix within mathematical computations and vector space analysis.
Recognizing these properties helps when delving into more complex matrix operations and understanding how matrices are organized and manipulated.
Two-Dimensional Matrices
Two-dimensional matrices, such as the matrix \(M_{23}\), are a fundamental part of linear algebra. They are often used in solving systems of equations, computer graphics, and data organization. Here are some features of two-dimensional matrices:
  • They are rectangular arrays and consist of numbers or expressions arranged in rows and columns.
  • The first number in the dimension \((m \times n)\) refers to the number of rows, while the second denotes the number of columns.
  • The total number of elements in a two-dimensional matrix is the product of its row and column count. For \(M_{23}\), that would be 6 elements in total.

These matrices provide a structured way to handle complex mathematical computations and are foundational in understanding matrix algebra.
Dimension Calculation
Calculating the dimension of a matrix space is simple once you understand the layout of the matrix. The dimensions essentially tell you the "size" or "capacity" of the matrix space.
  • For a matrix \(M_{mn}\), where \(m\) represents the number of rows and \(n\) is the number of columns, the dimension of the vector space is calculated as \(m \times n\).
  • Thus, for \(M_{23}\), the dimension would be \(2 \times 3 = 6\). This means this vector space is capable of containing six elements, organized in a 2 by 3 order.
  • This concept is invaluable for tasks like determining the rank of a matrix, performing matrix algebra, and in understanding theoretical aspects of vector spaces.

Being able to derive the dimension swiftly is a fundamental skill in matrix algebra that facilitates further exploration and manipulation of matrix data.

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Most popular questions from this chapter

Determine which functions are solutions of the linear differential equation. \(y^{\prime \prime}+y=0\) (a) \(e^{x}\) (b) \(\sin x\) (c) \(\cos x\) (d) \(\sin x-\cos x\)

Determine which functions are solutions of the linear differential equation. \(y^{\prime}-2 x y=0\) (a) \(y=3 e^{x^{2}}\) (b) \(y=x e^{x^{2}}\) (c) \(y=x^{2} e^{x}\) (d) \(y=x e^{-x}\)

Find the transition matrix from \(B\) to \(B^{\prime}\) by hand $$B=\\{(1,0),(0,1)\\}, B^{\prime}=\\{(2,4),(1,3)\\}$$

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Identify and sketch the graph. $$4 x^{2}+y^{2}-8 x+3=0$$
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