Question

Consider the linear transformations \(V \stackrel{T}{\rightarrow} W \stackrel{R}{\rightarrow} U\). a. Show that ker \(T \subseteq \operatorname{ker} R T\). b. Show that \(\operatorname{im} R T \subseteq \operatorname{im} R\).

Short answer

For part (a), ker \(T \subseteq \operatorname{ker}(RT)\). For part (b), \(\operatorname{im}(RT) \subseteq \operatorname{im}(R)\).

Step-by-step solution

Understanding the Kernels

Recall that \(\operatorname{ker}(T) = \{ v \in V : T(v) = 0_W \}\) and \(\operatorname{ker}(RT) = \{ v \in V : RT(v) = 0_U \}\). We need to show if \(@ v \in \operatorname{ker}(T), v \in \operatorname{ker}(RT)\).

Applying T and R on the Kernel

Take any element \(@ v \in \operatorname{ker}(T)\), which means \(@ T(v) = 0_W\). Compute \(@ RT(v) = R(T(v)) = R(0_W) = 0_U\). Therefore, \(@ v \in \operatorname{ker}(RT)\).

Concluding Ker T Subset

Since any element in \(@ \operatorname{ker}(T)\) also belongs to \(@ \operatorname{ker}(RT)\), it follows that \(@ \operatorname{ker}(T) \subseteq \operatorname{ker}(RT)\).

Understanding Image Transformations

Recall that \(@ \operatorname{im}(RT) = \{ RT(v) \in U : v \in V \}\) and \(@ \operatorname{im}(R) = \{ R(w) \in U : w \in W \}\). We aim to show that every element of \(@ \operatorname{im}(RT)\) is also in \(@ \operatorname{im}(R)\).

Examine Any Element in Image RT

For element \(@ u \in \operatorname{im}(RT), u = RT(v)\) for some \(@ v \in V\). This implies \(@ u = R(T(v)) \). Since \(@ T(v) \in W, u \in \operatorname{im}(R)\).

Concluding Image Subset

Thus, every element of \(@ \operatorname{im}(RT)\) is indeed an element of \(@ \operatorname{im}(R)\), hence \(@ \operatorname{im}(RT) \subseteq \operatorname{im}(R)\).

Key concepts

Kernel of a Transformation

The kernel of a transformation, often abbreviated as "ker", is a fundamental concept in linear algebra, specifically in the study of linear transformations between vector spaces. For a linear transformation \( T: V \rightarrow W \), the kernel is the set of all vectors in the domain \( V \) that are mapped to the zero vector in the codomain \( W \). Mathematically, it's expressed as \( \operatorname{ker}(T) = \{ v \in V : T(v) = 0_W \} \). This means that if you apply the transformation \( T \) to a vector \( v \) in the kernel, the result will be the zero vector of \( W \).

In terms of subset relationships, a pivotal property of the kernel is that it tells us about the injectivity of a transformation. If the only vector in \( \operatorname{ker}(T) \) is the zero vector, then \( T \) is injective, meaning no two different vectors in the domain are mapped to the same vector in the codomain.

For the exercise given, when considering the transformations from \( V \) to \( U \) via \( W \) using \( T \) and \( R \), we showed that the kernel of \( T \) is a subset of the kernel of the composition \( RT \). This means that vectors making \( T(v) \) zero will also make \( R(T(v)) \) zero, solidifying the kernel relationship between these composed transformations.

Image of a Transformation

The image of a transformation is another key concept referring to the range or output of a linear transformation. For a transformation \( T: V \rightarrow W \), the image is the set of all vectors in \( W \) that can be obtained by applying \( T \) to some vector in \( V \). More formally, \( \operatorname{im}(T) = \{ T(v) \in W : v \in V \} \). The image tells us about the surjectivity of the transformation, in that for surjection, the image of \( T \) should cover the entire codomain \( W \).

When analyzing the composite transformation \( RT \), the exercise asked us to confirm that the image of \( RT \) is a subset of the image of \( R \). This means every vector produced by \( RT \) in the codomain \( U \) is also one that could be produced by \( R \). Here, since the operation \( R(T(v)) \) implies the prerequisite \( T(v) \) belongs to \( W \), any result from \( RT \) is naturally contained within the results \( R \) can produce, emphasizing that composite transformations don't extend beyond the image of the following transformation.

Subset Relationships

Subset relationships in transformations stem from understanding how elements behave within sets defined by linear mappings. When we discussed the kernel and image, we explored how certain subsets—whether of the domain or codomain—relate to each other across transformations. The inclusion \( \operatorname{ker}(T) \subseteq \operatorname{ker}(RT) \) suggests all elements making \( T \) map to zero will do the same when \( T \) is composed with \( R \). Similarly, \( \operatorname{im}(RT) \subseteq \operatorname{im}(R) \) tells us that all outputs formed by the full transformation \( RT \) are within those selectable by \( R \).

In the language of vector spaces and transformations:

• "Ker" inclusion emphasizes that additional transformations don't expand initial nullifying vectors.


• "Im" inclusion illustrates how composed transformations are restricted by their subsequent application.

Such subset relations are essential in predicting the results and behavior of complex transformations, ensuring that linear mappings behave predictably in combined forms.