# Vector Spaces & Subspaces, Linear Algebra

• kash25
In summary, the conversation discusses the concept of a subspace of a vector space and how it relates to a given vector x. The conversation mentions the Subspace Test and suggests starting with the forward direction to show that T is a subspace if x is in U.
kash25

## Homework Statement

Let V be a vector space and U a subspace of V . For a given x ∈ V , deﬁne T=
{x + u | u ∈ U }. Show that T is a subspace of V if and only if x ∈ U .

## Homework Equations

Subspace Test:
1: The 0 vector of V is included in T.
2: T is closed under vector addition
3: T is closed under scalar multiplication

## The Attempt at a Solution

I do not know how to show this...

You're not trying very hard. Start with the forward direction. Show T is a subspace if x is in U. Try showing in this case T=U.

## 1. What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and a set of operations that can be performed on those vectors. These operations include addition and scalar multiplication, and they must follow certain rules in order for the set to be considered a vector space.

## 2. What is a subspace?

A subspace is a subset of a vector space that also satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, and it contains the zero vector.

## 3. How do you determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if none of the vectors in the set can be expressed as a linear combination of the other vectors. In other words, no vector in the set is redundant and adding or removing a vector would change the span of the set. This can be determined by setting up a system of equations and solving for the coefficients.

## 4. What is the difference between a basis and a spanning set?

A basis is a set of linearly independent vectors that span the entire vector space, meaning that any vector in the vector space can be written as a linear combination of the basis vectors. A spanning set is a set of vectors that can generate the entire vector space, but they may not necessarily be linearly independent.

## 5. Can a vector space have more than one basis?

Yes, a vector space can have many different bases. In fact, any linearly independent set of vectors that spans the vector space can be considered a basis. However, all bases for a particular vector space must have the same number of vectors, which is known as the dimension of the vector space.

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