# Finding a Basis for Subspace V Spanned by (1 1 2) etc.

• Dell
In summary, in order to find a basis for subspace V, which is spanned by the vectors (1 1 2), (2 -1 1), and (1 -2 -1), the person put them into a matrix system and performed elementary operations to get the vectors (1 1 2), (0 -3 -3), and (0 -3 -3), indicating that dimV=2. However, there are infinite possible bases for this vector space, so there is no "correct" answer for providing a basis.
Dell
if i am asked to fin a basis for the subspace V, which is spanned by ( 1 1 2) ( 2 -1 1) (1 -2 -1)...

i put them into a matrix system
1 1 2
2 -1 1
1 -2 -1
now after performing elementart operations i get
1 1 2
0 -3 -3
0 -3 -3
so since R3 and R2 are the same, dimV=2, my question is if i am asked to give a basis, should i give ( 1 1 2) (0 -3 -3) or should i return to the original vectors given and answer (1 1 2 ) (2 -1 1) or perhaps something else,, is ther a more correct answer if i am asked to give a basis

There are an infinite number of possible bases for every vector space. They are all equally valid. Selecting one over another does not matter.

## What is a subspace?

A subspace is a subset of a vector space that satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector. In simpler terms, it is a set of vectors that can be combined and stretched to form all other vectors within that space.

## What does it mean for a subspace to be spanned by a set of vectors?

A subspace is said to be spanned by a set of vectors if all vectors within that subspace can be formed by taking linear combinations of those vectors. In other words, the set of vectors can be used to create all other vectors within that subspace.

## What is a basis for a subspace?

A basis for a subspace is a set of linearly independent vectors that span the subspace. This means that no vector in the set can be written as a combination of the other vectors, and together they can form all other vectors within the subspace.

## How do you find a basis for a subspace?

To find a basis for a subspace, you can use the process of Gaussian elimination to reduce the set of vectors to its simplest form. The resulting vectors will be linearly independent and span the subspace.

## Why is finding a basis for a subspace important?

Finding a basis for a subspace is important because it allows us to understand the structure of the subspace and its relationships to other subspaces. It also enables us to perform calculations and operations within that subspace more efficiently.

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