Sums and intersections of subspaces worksheets for university Linear Algebra students

📚 Topic Summary

In linear algebra, understanding how subspaces interact is crucial. The sum of subspaces combines elements from each subspace to create a larger space. Formally, if $U$ and $W$ are subspaces of a vector space $V$, their sum, denoted $U + W$, is the set of all vectors $u + w$ where $u \in U$ and $w \in W$. The intersection of subspaces, denoted $U \cap W$, contains vectors that belong to both $U$ and $W$. Understanding these concepts allows us to analyze the structure of vector spaces more effectively. These are important concepts to grasp to further advance in your linear algebra course.

🧮 Part A: Vocabulary

Match the terms with their definitions:

1. Term: Subspace Sum
2. Term: Subspace Intersection
3. Term: Vector Space
4. Term: Linear Combination
5. Term: Span

1. Definition: The set of all linear combinations of a set of vectors.
2. Definition: A collection of objects that behave like vectors.
3. Definition: The set of all vectors that belong to both subspaces.
4. Definition: A subset of a vector space that is closed under addition and scalar multiplication.
5. Definition: The set of all vectors formed by adding vectors from each subspace.

✍️ Part B: Fill in the Blanks

Complete the following paragraph with the correct terms:

The _______ of two subspaces, $U$ and $W$, contains vectors that are present in both $U$ and $W$. On the other hand, the _______, denoted $U + W$, consists of all vectors that can be written as the sum of a vector from $U$ and a vector from $W$. Showing that the _______ of two subspaces is also a subspace is an important exercise.

🤔 Part C: Critical Thinking

Explain, using examples, how the dimension of the sum of two subspaces relates to the dimensions of the individual subspaces and their intersection.