Question

Let \(R\) and \(S\) be matrix transformations \(\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) induced by matrices \(A\) and \(B\) respectively. In each case, show that \(T\) is a matrix transformation and describe its matrix in terms of \(A\) and \(B\). a. \(T(\mathbf{x})=R(\mathbf{x})+S(\mathbf{x})\) for all \(\mathbf{x}\) in \(\mathbb{R}^{n}\). b. \(T(\mathbf{x})=a R(\mathbf{x})\) for all \(\mathbf{x}\) in \(\mathbb{R}^{n}\) (where \(a\) is a fixed real number).

Short answer

a. The matrix for \( T \) is \( A + B \); b. The matrix for \( T \) is \( aA \).

Step-by-step solution

Understand Matrix Transformations

Matrix transformations are functions that take a vector from one space and output a vector in another, using a matrix to map this transformation. For instance, matrix transformation \( R \) maps \( \mathbf{x} \) from \( \mathbb{R}^n \) to \( \mathbb{R}^m \) using matrix \( A \): \( R(\mathbf{x}) = A\mathbf{x} \). Similarly, \( S(\mathbf{x}) = B\mathbf{x} \).

Establish Matrix for T(x) = R(x) + S(x)

Transformation \( T(\mathbf{x}) = R(\mathbf{x}) + S(\mathbf{x}) \) implies \( T(\mathbf{x}) = A\mathbf{x} + B\mathbf{x} = (A + B)\mathbf{x} \). Since matrix addition is well-defined, \( T(\mathbf{x}) \) is a linear transformation and its matrix is \( A + B \).

Confirm Linear Transformation Properties

To confirm \( T(\mathbf{x}) = R(\mathbf{x}) + S(\mathbf{x}) \) is a linear transformation, it must satisfy linearity, i.e., \( T(c\mathbf{x} + d\mathbf{y}) = cT(\mathbf{x}) + dT(\mathbf{y}) \). \( A + B \) satisfies this since both \( A \) and \( B \) individually satisfy linearity.

Establish Matrix for T(x)=aR(x)

For \( T(\mathbf{x}) = a R(\mathbf{x}) \), we have: \( T(\mathbf{x}) = a A\mathbf{x} = (aA)\mathbf{x} \). Thus, the transformation \( T \) is also linear and induced by the matrix \( aA \).

Confirm Scalars Maintain Linearity

Check \( T(\mathbf{x}) = a R(\mathbf{x}) \) for linearity: \( T(c\mathbf{x} + d\mathbf{y}) = a A(c\mathbf{x} + d\mathbf{y}) = ac A\mathbf{x} + ad A\mathbf{y} = c(aA)\mathbf{x} + d(aA)\mathbf{y} \). Linearity is maintained.

Key concepts

Linear Algebra

Linear algebra is a branch of mathematics that centers around vector spaces and linear mappings between these spaces. One of the most powerful applications of linear algebra is the study and use of matrices as tools for transforming these spaces.

Understanding matrix transformations is crucial in linear algebra because any linear transformation between finite-dimensional vector spaces can be represented by a matrix. This makes linear algebra not only theoretical but also very applicable, providing a concrete way to manipulate data through transformations.

In our exercise, when considering transformations from a vector space \( \mathbb{R}^{n} \) to another vector space \( \mathbb{R}^{m} \), matrices \( A \) and \( B \) act as the mechanisms that convert input vectors to output vectors. This transformation can be expressed in the form of functions \( R(\mathbf{x}) \) and \( S(\mathbf{x}) \). Each matrix is like a map that transforms vectors from their initial state in \( \mathbb{R}^{n} \) to their new state in \( \mathbb{R}^{m} \).
Providing a solid understanding of these transformations allows us to derive results about the structure and functionality of systems in physical sciences, engineering, computer graphics, and more. This connection between theoretical concepts and practical applications shows the robustness of linear algebra in solving real-world problems.

Matrix Operations

Matrix operations are fundamental tools for performing manipulation on matrices to achieve various transformations.

In our exercise, we encounter two basic matrix operations: addition and scalar multiplication.
- **Matrix Addition**: Two matrices can only be added if they are of the same dimensions. If matrices \( A \) and \( B \) represent transformations \( R \) and \( S \) respectively, adding them results in a new transformation \( T(\mathbf{x}) = (A + B)\mathbf{x} \). This operation applies component-wise addition to each pair of corresponding elements from the matrices.

- **Scalar Multiplication**: When a matrix \( A \) is multiplied by a scalar \( a \), every element of the matrix is multiplied by \( a \). In the case \( T(\mathbf{x}) = a R(\mathbf{x}) \), this operation results in \( T(\mathbf{x}) = (aA)\mathbf{x} \). Scalar multiplication is essential for scaling transformations, and it preserves the dimensions of the matrix while adjusting the transformation effect based on the scalar.
Matrix operations must maintain the properties of linearity, ensuring that the transformations they represent can be combined, scaled, and manipulated while retaining linear characteristics. Learning these operations is key to understanding more complex transformations and systems.

Linearity Properties

Linearity is a cornerstone property of transformations in linear algebra. It ensures that transformations respect both vector addition and scalar multiplication.

A transformation \( T \), such as the ones induced by matrices \( A \), \( B \), or combinations like \( A + B \) or \( aA \), is linear if it satisfies two conditions:

• **Additivity:** \( T(\mathbf{x} + \mathbf{y}) = T(\mathbf{x}) + T(\mathbf{y}) \) for any vectors \( \mathbf{x} \) and \( \mathbf{y} \).
• **Homogeneity:** \( T(a\mathbf{x}) = aT(\mathbf{x}) \) for any vector \( \mathbf{x} \) and scalar \( a \).

For both \( T(\mathbf{x}) = R(\mathbf{x}) + S(\mathbf{x}) \) and \( T(\mathbf{x}) = a R(\mathbf{x}) \), additivity and homogeneity hold true, confirming them as linear transformations.

When we verify these properties in our exercise, we affirm that matrix operations like addition and scalar multiplication do not violate linearity. This verification is crucial because many results in linear algebra depend on the assumption that the transformations involved are linear. Maintaining linearity ensures that solutions derived from these transformations are consistent and predictable.