Question

Prove: If \(\mathbf{L}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is a linear transformation, then $$ \mathbf{L}\left(a_{1} \mathbf{X}_{1}+a_{2} \mathbf{X}_{2}+\cdots+a_{k} \mathbf{X}_{k}\right)=a_{1} \mathbf{L}\left(\mathbf{X}_{1}\right)+a_{2} \mathbf{L}\left(\mathbf{X}_{2}\right)+\cdots+a_{k} \mathbf{L}\left(\mathbf{X}_{k}\right) $$ if \(\mathbf{X}_{1}, \mathbf{X}_{2}, \ldots, \mathbf{X}_{k}\) are in \(\mathbb{R}^{n}\) and \(a_{1}, a_{2}, \ldots . a_{k}\) are real numbers.

Short answer

Question: Prove that a given linear transformation equation holds true for provided vectors and scalars using the properties of additivity and homogeneity. Answer: To prove the given linear transformation equation, we can follow these steps: 1. Recall the properties of linear transformations, which are additivity and homogeneity. 2. Break down the given equation into multiple terms. 3. Apply the additivity property to simplify the equation. 4. Apply the homogeneity property to each term of the equation. If the terms on both sides of the equation are equal after applying these properties, the given linear transformation equation is true for the provided vectors and scalars.

Step-by-step solution

Recall the properties of linear transformations

Linear transformations have two properties, which are as follows: 1. Additivity: \(\mathbf{L}(\mathbf{X}+\mathbf{Y})=\mathbf{L}(\mathbf{X})+\mathbf{L}(\mathbf{Y})\) 2. Homogeneity: \(\mathbf{L}(a\mathbf{X})=a\mathbf{L}(\mathbf{X})\) We will apply these two properties to simplify the given equation and prove the statement.

Break down the given equation

Initially, we have the following equation: $$ \mathbf{L}\left(a_{1} \mathbf{X}_{1}+a_{2} \mathbf{X}_{2}+\cdots+a_{k} \mathbf{X}_{k}\right)=a_{1} \mathbf{L}\left(\mathbf{X}_{1}\right)+a_{2} \mathbf{L}\left(\mathbf{X}_{2}\right)+\cdots+a_{k} \mathbf{L}\left(\mathbf{X}_{k}\right). $$ We can break down the equation into multiple terms: $$ \mathbf{L}(\sum_{i=1}^k a_i \mathbf{X}_i) = \sum_{i=1}^k a_i \mathbf{L}(\mathbf{X}_i) $$

Apply the additivity property

By applying the additivity property repeatedly, we get: $$ \mathbf{L}(a_1 \mathbf{X}_1 + a_2 \mathbf{X}_2 + \cdots + a_k \mathbf{X}_k) = \mathbf{L}(a_1\mathbf{X}_1) + \mathbf{L}(a_2\mathbf{X}_2) + \cdots + \mathbf{L}(a_k\mathbf{X}_k) $$

Apply the homogeneity property

Applying the homogeneity property for each term, we get: $$ a_1\mathbf{L}(\mathbf{X}_1)+a_2\mathbf{L}(\mathbf{X}_2)+\cdots+a_k\mathbf{L}(\mathbf{X}_k) = a_1\mathbf{L}(\mathbf{X}_1)+a_2\mathbf{L}(\mathbf{X}_2)+\cdots+a_k\mathbf{L}(\mathbf{X}_k) $$ Since the terms on both sides of the equation are equal, we can conclude that the given linear transformation equation is true for vectors \(\mathbf{X}_1\) to \(\mathbf{X}_k\) and scalars \(a_1\) to \(a_k\). Hence, the proof is complete.

Key concepts

Real Analysis

Real analysis is a branch of mathematical analysis dealing with real numbers and real-valued functions. At its core, it involves the rigorous study of limits, continuity, differentiation, integration, and sequences of real numbers. One of the fundamental principles encountered in real analysis is the concept of a linear transformation.

A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The exercise presented gives us a scenario where a function from an -dimensional real number space to an m-dimensional real number space is being studied for its linearity. In real analysis, such a proof is crucial because it establishes that certain functions we work with behave predictably under the algebra of vectors, thereby allowing us to use these functions to explore more complex structures and relationships in mathematics.

When proving that a given function is a linear transformation in real analysis, we primarily focus on two attributes: additivity and homogeneity. The exercise uses these characteristics to demonstrate the central idea that linear transformations can be broken down into simpler operations which adhere to these two properties, a fundamental aspect in real analysis that aids in understanding more complicated mathematical concepts.

Properties of Linear Transformations

Understanding the properties of linear transformations is critical for grasping the behavior of these functions within a vector space. According to the exercise, a linear transformation must satisfy two main properties:

1. Additivity: This property states that the transformation of the sum of two vectors is the same as the sum of their respective transformations. Mathematically, for vectors X and Y, it is expressed as \(\mathbf{L}(\mathbf{X}+\mathbf{Y})=\mathbf{L}(\mathbf{X})+\mathbf{L}(\mathbf{Y})\).
2. Homogeneity: This property specifies that the transformation of a scalar multiple of a vector is the same as the scalar multiple of the transformed vector. In notation for a vector X and a scalar a, this reads \(\mathbf{L}(a\mathbf{X})=a\mathbf{L}(\mathbf{X})\).

The solution steps through the process of verifying these properties for a more complex scenario involving a combination of multiple vectors and scalars. By applying additivity and homogeneity sequentially, we are able to decompose and rebuild the original transformation equation, ultimately proving the exercise's statement. This methodical breakdown not only solves the problem at hand but serves as a model for tackling similar proofs in linear algebra and beyond.

Vector Space

A vector space is a fundamental mathematical construct in fields such as linear algebra and real analysis. It consists of a collection of elements called vectors that can be added together and multiplied by scalars (real or complex numbers). For a set to qualify as a vector space, it must satisfy certain axioms such as the existence of a zero vector, the ability for each vector to have an additive inverse, and the distributive property for vector addition and scalar multiplication.

In the context of the given exercise, the vectors \(\mathbf{X}_{1}, \mathbf{X}_{2}, \ldots, \mathbf{X}_{k}\) are all elements of an -dimensional vector space over the real numbers, commonly denoted \(\mathbb{R}^{n}\). The exercise highlights the significant role vector spaces play in defining linear transformations. Here, linearity ensures that any linear combination of vectors in the space (a sum weighted by scalars, as seen in the proof) results in another vector in the same space after the transformation.

Recognizing this concept is essential for palpating the structure and constraints of various mathematical spaces, particularly when dealing with higher dimensions and more abstract forms, which regularly feature in real analysis and other advanced mathematical disciplines.