# Vector spaces, subspaces, subsets, intersections

• karnten07
In summary, we are trying to show that for a vector space V over a field F, if X, Y, and Z are subspaces of V such that X is a subset of Y, then the intersection of Y and the sum of X and Z is equal to the sum of X and the intersection of Y and Z. This can be shown by using the fact that every element in the left-hand side is contained in the right-hand side, and vice versa. In other words, if v is in Y and can be written as au + bw, where u is in X and w is in Z, then it can also be written as au + bw', where u is in X and w' is in Y and Z.
karnten07

## Homework Statement

Let V be a vector space over a field F and let X, Y and Z be a subspaces of V such that X$$\subseteq$$Y. Show that Y$$\cap$$(X+Z) = X + (Y$$\cap$$Z). (Hint. Show that every element of the LHS is contained on the RHS and vice versa.)

## The Attempt at a Solution

Can anyone get me started on this one?

Suppose that you have an element $p \in Y \cap (X + Z)$. Then $p \in Y$ and $p \in X + Z$. The latter means we can write $p = x + z$ with $x \in X, z \in Z$. Now do you see how you can also write it as $x' + y'$ with $x' \in X, y' \in Y \cap Z$?

karnten07 said:

## Homework Statement

Let V be a vector space over a field F and let X, Y and Z be a subspaces of V such that X$$\subseteq$$Y. Show that Y$$\cap$$(X+Z) = X + (Y$$\cap$$Z). (Hint. Show that every element of the LHS is contained on the RHS and vice versa.)

## The Attempt at a Solution

Can anyone get me started on this one?
The hint looks like all you need. Have you tried that at all? Suppose v is in $Y\cap(X+Z)$. That means it is in y and it can be written v= au+ bw where u is in X and w is in Z. Now you need to show that v is in $X+ (Y\cap Z)$. That is, that it can be written in the form au+ bw where u is in X and w is in $Y\cap Z$. Once you have done that turn it around: if v is in $X+ (Y\cap Z)$, can you show that it must be in $Y\cap (X+ Z)$?

## 1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects called vectors, along with operations of addition and scalar multiplication. These operations follow certain properties, such as closure, associativity, and distributivity, and allow for the manipulation of vectors within the space.

## 2. How is a subspace defined?

A subspace is a subset of a vector space that also satisfies the properties of a vector space. This means that it must contain the zero vector, be closed under addition and scalar multiplication, and satisfy the other properties of a vector space. Subspaces are useful for breaking down a larger vector space into smaller, more manageable parts.

## 3. What is the difference between a subset and a subspace?

A subset is simply a collection of elements from a larger set, while a subspace is a subset that satisfies the properties of a vector space. This means that a subspace is a more specific type of subset, with additional requirements for the elements within it.

## 4. How do I find the intersection of two subspaces?

The intersection of two subspaces is the set of all elements that are common to both subspaces. To find the intersection, you can use the equations of the two subspaces and solve for the variables. The resulting solution will be the set of elements that are in both subspaces.

## 5. Can a vector space have multiple subspaces?

Yes, a vector space can have multiple subspaces. In fact, every vector space has at least two subspaces - the entire vector space itself and the subspace consisting of only the zero vector. Additionally, there can be an infinite number of subspaces within a vector space, as long as they satisfy the properties of a vector space.

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