Question

Suppose \(\rho: M \rightarrow M^{\prime}\) is an \(R\) -linear map. Show that if \(\left\\{\alpha_{i}\right\\}_{i=1}^{n}\) is a linearly dependent family of elements of \(M,\) then \(\left\\{\rho\left(\alpha_{i}\right)\right\\}_{i=1}^{n}\) is also linearly dependent.

Short answer

Question: Given an R-linear map ρ: M → M' and a linearly dependent family \(\left\\{\alpha_{i}\right\\}_{i=1}^{n}\) of elements from M, show that the family of elements \(\left\\{\rho(\alpha_{i})\right\\}_{i=1}^{n}\) after applying the R-linear map ρ is also linearly dependent. Answer: Since \(\left\\{\alpha_{i}\right\\}_{i=1}^{n}\) is a linearly dependent family of elements from M, there exist scalars \(r_i \in R\), not all zero, such that \(\sum_{i=1}^{n} r_i \alpha_i = 0\). By applying the R-linear map ρ and using its properties, we find that \(\sum_{i=1}^{n} r_i \rho(\alpha_i) = \rho(0)\). This shows that the family of elements \(\left\\{\rho(\alpha_{i})\right\\}_{i=1}^{n}\) is also linearly dependent.

Step-by-step solution

Definition of Linearly Dependent Family

Recall that a family of elements \(\left\\{\alpha_{i}\right\\}_{i=1}^{n}\) from \(M\) is linearly dependent if there exists scalars \(r_i \in R\), not all zero, such that \(\sum_{i=1}^{n} r_i \alpha_i = 0\).

Definition of R-Linear Map

An R-linear map ρ: M → M' satisfies the following properties for all \(m_1, m_2 \in M\) and \(r \in R\): 1. \(\rho(m_1 + m_2) = \rho(m_1) + \rho(m_2)\) 2. \(\rho(r\cdot m_1) = r\cdot \rho(m_1)\)

Applying the R-Linear Map to a Linearly Dependent Family

We know that \(\left\\{\alpha_{i}\right\\}_{i=1}^{n}\) is a linearly dependent family of elements of \(M\). So, there exist scalars \(r_i \in R\), not all zero, such that \(\sum_{i=1}^{n} r_i \alpha_i = 0\). Now, let's apply the R-linear map ρ to this equation: $$\rho\left(\sum_{i=1}^{n} r_i \alpha_i\right) = \rho(0)$$

Using Properties of R-Linear Map

Using the properties of an R-linear map, we can rewrite the left-hand side of the equation in Step 3 as follows: $$\sum_{i=1}^{n} r_i \rho(\alpha_i) = \rho(0)$$

Proving the Resulting Family is Linearly Dependent

We have shown that there exist scalars \(r_i \in R\), not all zero, such that \(\sum_{i=1}^{n} r_i \rho(\alpha_i) = \rho(0)\). By the definition of linearly dependent family, this implies that the family of elements \(\left\\{\rho(\alpha_{i})\right\\}_{i=1}^{n}\) is also linearly dependent. Thus, we have proven the statement.

Key concepts

R-linear map

An \(R\)-linear map is a fundamental concept in linear algebra that deals with functions between two linear spaces that preserve the structure of addition and scalar multiplication. To distinguish it from linear maps over other fields or rings, we specify \(R\)-linear to indicate that the scalars involved are from a ring \(R\).

The essence of an \(R\)-linear map \(\rho: M \rightarrow M'\) is captured by two main properties: First, if you take two elements \(m_1\) and \(m_2\) from \(M\), the map respects their addition. This means when you apply \(\rho\) to the sum of \(m_1\) and \(m_2\), it's the same as applying \(\rho\) to each one individually and then adding the results. This property can be formally expressed as \(\rho(m_1 + m_2) = \rho(m_1) + \rho(m_2)\).

Second, the map also respects scalar multiplication. When you multiply an element \(m_1\) in \(M\) by a scalar \(r\) from \(R\) and then apply \(\rho\), you will get the same result as applying \(\rho\) to \(m_1\) first and then multiplying by \(r\). Mathematically, this is shown as \(\rho(r\cdot m_1) = r\cdot \rho(m_1)\).

These two properties ensure that the structure of the linear space is maintained under the map, making \(\rho\) a tool essential for transitioning between spaces while preserving linear relationships.

Linearly dependent family

The concept of a linearly dependent family is pivotal to understanding the geometry and algebra of vector spaces. A family of elements \(\left\{\alpha_{i}\right\}_{i=1}^{n}\) from a module \(M\) is considered linearly dependent if there is a way to express the zero element of that module as a non-trivial linear combination of the elements in the family. In more plain terms, if you can find some scalars \(r_i\), where not all of them are zero, such that when you multiply each family member by its corresponding scalar and add up the results, you get zero, the family is linearly dependent. This is represented by the equation \(\sum_{i=1}^{n} r_i \alpha_i = 0\).

It's important to note that this dependency implies that there is a certain redundancy in the family of elements: some elements can be derived from others. This contrasts with a linearly independent family where no such combination exists except when all scalars are zero. Understanding whether a set of vectors is linearly dependent or independent is crucial for various tasks, such as dimensionality reduction, solving systems of linear equations, and determining spans and bases of vector spaces.

Properties of linear maps

The properties of linear maps are the backbone of much of linear algebra theory and application. These properties define how a linear map interacts with the elements of the spaces it connects. Specifically, any linear map obeys two core rules:

Additivity: If you take any two elements from the source space and add them together, applying the linear map to this sum is the same as applying the map to each element separately and then adding those results. This property ensures that the concept of 'direction' is preserved under the map.

Homogeneity: When you scale an element by a scalar (from the ring or field over which the space is defined) and then apply the linear map, it's equivalent to applying the map first and then scaling the result. This ensures that 'scaling' or 'stretching' factors are maintained.

These properties allow us to perform algebraic manipulations that are predictable and consistent, making linear maps incredibly valuable for tasks like transformations between coordinate systems, solving differential equations, and even in computer graphics for rendering transformations. For instance, when proving results like the one in our original exercise, the adherence to these properties guarantees that the phenomena of linear dependence or independence of elements are preserved through the mapping process.