Final Answer:
a. The dimension of V is m.
b. The dimension of W is n.
c. The dimension of X is p.
d. The dimension of Y is q.
e. The dimension of Z is r.
Step-by-step explanation:
In linear algebra, the dimension of a subspace is a fundamental concept that represents the number of linearly independent vectors needed to span that subspace. Let's denote the subspaces as V, W, X, Y, and Z. If V has m linearly independent vectors, then the dimension of Z is m. Similarly, if W has n linearly independent vectors, the dimension of W is n.
To calculate the dimension of a subspace, one can use methods like row reduction, rank-nullity theorem, or inspecting the span of vectors in the subspace. For example, if X has p linearly independent vectors, its dimension is p. The same reasoning applies to Y and Z—if Y has q linearly independent vectors, its dimension is q, and if Z has r linearly independent vectors, its dimension is r.
Understanding the dimension of subspaces is crucial for various applications, including solving systems of linear equations, studying linear transformations, and comprehending the structure of vector spaces. The concept of dimension provides insights into the underlying geometry and structure of vector spaces and their subspaces.