# What are the subspaces of ℝ, ℝ^2, and ℝ^3?

• mtayab1994
In summary, a subspace in linear algebra is a subset of a vector space that satisfies the three fundamental properties. It is a smaller set of vectors that can still be added and multiplied by scalars within the same vector space. A subspace is different from a span, which is the set of all possible linear combinations of a given set of vectors. To determine if a set of vectors is a subspace, you need to check if it satisfies the three fundamental properties and is closed under linear combinations. The dimension of a subspace is the minimum number of vectors needed to span the subspace and is always less than or equal to the dimension of the entire vector space. A subspace can be a line or a point, as long as
mtayab1994

## Homework Statement

Find all subspaces of the vector spaces: (ℝ+,.) , (ℝ^2 +,.) , (ℝ^3 +,.)

## The Attempt at a Solution

For ℝ the only subspace i can think of is {0}

For ℝ^2 if found {0} R^2 itself and any set of the form L=cu for u≠0.

For ℝ^3 if found those of R^2 plus R^3 . Are they correct?

For the second case, you found that L = cu, where c is some given constant, including 0, defines a subspace. Find a similar equation for the next case, and think how you can generalize that.

## What is a subspace in linear algebra?

A subspace in linear algebra is a subset of a vector space that satisfies the three fundamental properties: closure under vector addition, closure under scalar multiplication, and containing the zero vector. In other words, a subspace is a smaller set of vectors that can still be added and multiplied by scalars to produce vectors within the same vector space.

## What is the difference between a subspace and a span?

A subspace is a subset of a vector space that satisfies the three fundamental properties, while a span is the set of all possible linear combinations of a given set of vectors. In other words, a span is a subspace if and only if the span includes the zero vector and is closed under vector addition and scalar multiplication.

## How do you determine if a set of vectors is a subspace?

To determine if a set of vectors is a subspace, you need to check if it satisfies the three fundamental properties: closure under vector addition, closure under scalar multiplication, and containing the zero vector. You can also check if the set is closed under linear combinations, meaning that any linear combination of vectors in the set will still be in the set.

## What is the dimension of a subspace?

The dimension of a subspace is the number of vectors in a basis for that subspace. In other words, it is the minimum number of vectors needed to span the subspace. The dimension of a subspace is always less than or equal to the dimension of the entire vector space.

## Can a subspace be a line or a point?

Yes, a subspace can be a line or a point, as long as it satisfies the three fundamental properties of a subspace. For example, a line through the origin in a two-dimensional vector space is a subspace, and a single point in a three-dimensional vector space is also a subspace.

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