Problem 17 - Calculus for Biology and Medicine

Question

Suppose that \(A\) and \(B\) are \(m \times n\) matrices. Show that $$ (A+B)^{\prime}=A^{\prime}+B^{\prime} $$

Step-by-Step Solution

Answer

The transpose of the sum is equal to the sum of the transposes: \((A+B)' = A' + B'\).

Step 1: Understand the Problem

We are given two matrices, \(A\) and \(B\), both of size \(m \times n\). We need to show that the transpose (denoted as \(^{\prime}\)) of their sum \((A+B)\) is equal to the sum of their transposes, \(A^{\prime} + B^{\prime}\).

Step 2: Apply the Definition of Transpose

The transpose of a matrix, in general, is obtained by switching its rows and columns. Thus, for any matrix \(M\) of size \(m \times n\), the transpose \(M^{\prime}\) will be of size \(n \times m\).

Step 3: Transpose the Sum (A+B)

Let's consider \((A + B)^{\prime}\). According to the transpose operation, we switch the rows and columns of the resulting matrix \((A + B)\), which is itself of size \(m \times n\).

Step 4: Use the Property of Each Element in the Transposed Matrix

For two matrices \(X\) and \(Y\), the sum \((X + Y)\) is computed element-wise: \((X + Y)_{ij} = X_{ij} + Y_{ij}\). When we take the transpose, we have: \[(A + B)^{\prime}_{ij} = (A + B)_{ji} = A_{ji} + B_{ji}\].

Step 5: Compare with A' + B'

For the matrices \(A\) and \(B\), their transposes are \(A^{\prime}\) and \(B^{\prime}\) respectively, where \(A^{\prime}_{ij} = A_{ji}\) and \(B^{\prime}_{ij} = B_{ji}\). Thus, \[A^{\prime}_{ij} + B^{\prime}_{ij} = A_{ji} + B_{ji}\].

Step 6: Conclude the Proof

Since \((A+B)^{\prime}_{ij} = A^{\prime}_{ij} + B^{\prime}_{ij}\) for each element \(i,j\), we conclude that: \[(A + B)^{\prime} = A^{\prime} + B^{\prime}\].

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra, allowing us to combine two matrices of the same dimensions into a single matrix. Imagine you have two matrices, say \( A \) and \( B \), each with dimensions \( m \times n \). To perform matrix addition, you simply add the corresponding elements from each matrix. This means that for each element located at the \( i \)-th row and \( j \)-th column in the resultant matrix, it's obtained by adding the numbers at the same position in matrices \( A \) and \( B \).

• For example, if \( A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} \) and \( B = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix} \), then their sum \( A + B = \begin{pmatrix} 1+5 & 2+6 \ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \ 10 & 12 \end{pmatrix} \).
• This operation is straightforward and can be thought of as overlapping two matrices and adding up the numbers that visually align.

This addition only works if both matrices have the same size, so you can't directly add a \( 2 \times 3 \) matrix to a \( 3 \times 2 \) matrix. Always make sure that the dimensions match before proceeding with matrix addition.

Linear Algebra

Linear Algebra is the branch of mathematics that focuses on the study of vectors, vector spaces (also known as linear spaces), linear transformations, and systems of linear equations. It's a central component of many scientific and engineering disciplines, providing the foundational framework for computations.
Among the key concepts you'll encounter in linear algebra are matrices and their operations, such as addition, scalar multiplication, and transposition. Matrices are powerful tools, not only in handling systems of equations but also in transforming geometric shapes and managing data in various dimensions.
Linear mappings, which translate to linear transformations in terms of matrices, are another fundamental aspect. They allow us to map elements from one vector space to another using linear functions. These transformations can be easily represented using matrices, making calculations more efficient, especially when dealing with transformations like rotations and reflections in graphics or data manipulation in machine learning tasks.

Elementary Matrix Operations

In linear algebra, understanding elementary matrix operations is crucial because they form the building blocks for more complex manipulations such as solving linear equations or performing matrix decompositions. The three primary operations include matrix addition, scalar multiplication, and matrix transposition.

• Matrix Addition: As previously explained, it's about adding corresponding elements of matrices of the same size.
• Scalar Multiplication: This involves multiplying every element of a matrix by a number (called a scalar). For a matrix \( A \) and a scalar \( c \), the result is a new matrix obtained by computing \( c \times A_{ij} \) for each element.
• Matrix Transposition: This operation switches the rows and columns of a matrix. Given a matrix \( A \) of size \( m \times n \), its transpose, \( A^{\prime} \), will be of size \( n \times m \). This swap is pivotal in proofs and solving many linear algebra problems, as seen in the provided exercise.

These operations are not only simple but also essential for practical applications. Whether it's data alignment in databases or transformations in quantum mechanics, elementary matrix operations offer a consistent method for handling and processing matrix-based data.