# Finding a Basis of W: Exploring Rows & Matrices

• Dell
In summary, to find a basis of W, a subspace of R4, you need to put the given vectors into a matrix and try to eliminate rows. If you cannot create a row of zeros, then the vectors are linearly independent and form the basis of their span. It does not matter if the matrix is "standing" or "lying down".
Dell
given these 3 vectors

w1=(1 0 1 -1)
w2=(2 1 2 -3)
w3=(3 1 1 -2)

and i am asked to find a basis of W, subspace of R4

to do this am i supposed to put these vectors into a matrix and try elliminate rows, leaving the basis being the rows i could not get rid of??

does it matter if i put them in a "standing" matrix "A" or a "lying down" matrix "At "

i tried this with an At like matrix, but couldn't elliminate any of them,? are they all 3 the basis or have i done something wrong?

If you can't create a row of zeros by elimination, then they are linearly independent and you need them all to form a basis of their span, yes.

## 1. What is a basis in linear algebra?

A basis in linear algebra is a set of vectors that span a vector space. This means that any vector in the vector space can be expressed as a linear combination of the basis vectors. A basis is also linearly independent, meaning none of the basis vectors can be written as a linear combination of the other basis vectors.

## 2. How do you find a basis of a vector space?

To find a basis of a vector space, you can use a variety of methods such as Gaussian elimination, row reduction, or finding the null space of a matrix. These methods involve manipulating the rows and columns of a matrix to find a set of linearly independent vectors.

## 3. What is the purpose of finding a basis?

Finding a basis is important in linear algebra because it helps us to understand the structure of a vector space and allows us to solve systems of linear equations. It also allows us to represent vectors and linear transformations in a more efficient and concise manner.

## 4. How is a basis related to rows and matrices?

A basis is related to rows and matrices through the concept of linear independence. The rows of a matrix can be thought of as vectors in a vector space and finding a basis of these rows means finding a set of linearly independent vectors that span the row space of the matrix.

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

Yes, a vector space can have more than one basis. In fact, there can be infinitely many bases for a vector space, as long as they all satisfy the properties of being linearly independent and spanning the vector space. This is because there are often many different ways to express a vector as a linear combination of basis vectors.

Replies
4
Views
1K
Replies
8
Views
1K
Replies
4
Views
842
Replies
14
Views
2K
Replies
2
Views
1K
Replies
15
Views
1K
Replies
1
Views
846
Replies
4
Views
2K
Replies
1
Views
1K
Replies
25
Views
3K