91Ó°ÊÓ

A vector space, also known as a linear space, is a fundamental concept in linear algebra. It is a collection of objects called vectors, which can be added together and multiplied by numbers, called scalars. In the context of our exercise, the vector space is denoted by the symbol \(V\), and it serves as a 'stage' upon which all mathematical objects (subspaces, vectors) act and interact.

For a better understanding, you can imagine a vector space as a vast field or arena where vectors can freely move around, combine, and stretch or shrink in size. The rules governing these movements and combinations are the principles of vector addition and scalar multiplication.

Closure under addition is one of the essential properties of a vector space or subspace. This property means that if you pick any two vectors from the vector space, their sum should also be a vector within the same space. For example, if \(u\) and \(v\) are vectors in a space \(W\), then the sum \(u + v\) must also be in \(W\).

Think of it like a school where students must wear a uniform. If two students wearing the school uniform stand together, they still represent the school. Similarly, in a vector space, when you combine vectors through addition, the result must fit in the 'uniform' of that space, showing it truly belongs there.

Closure under scalar multiplication is another property that a vector space must follow. It means that if you take any vector within the space and multiply it by a scalar - which is just a number like 2, -3, or 0.5 - the product is still a vector in the same space. In technical terms, if \(u\) is a vector in space \(W\) and \(c\) is a scalar, then the product \(cu\) remains within \(W\).

Using an earlier analogy, if our school has a rule that any book brought by a student automatically becomes school property, then the scalar is like the student bringing the book. Once the multiplication happens, the 'book' (the resulting vector) is part of the school's 'library' (the vector space).

Subspaces are essentially smaller 'rooms' within the vast 'house' of a vector space. Each subspace must meet certain criteria - they too must have closure under addition and scalar multiplication, while also containing the zero vector. These criteria ensure that every subspace is a valid vector space on its own right, just living within a larger one.

In the exercise, the intersection of multiple subspaces is being considered. This intersection is like the common area that exists in several rooms. If that area adheres to the house rules - non-emptiness (has the zero vector), closure under addition, and closure under scalar multiplication - then it is a legitimate 'room' or subspace within the larger 'house', or vector space. Our steps in the solution show that intersections indeed follow these rules, confirming their status as subspaces within the vector space \(V\).