Finding a Basis of W: Exploring Rows & Matrices

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".
  • #1
Dell
590
0
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?
 
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  • #2
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.
 

FAQ: Finding a Basis of W: Exploring Rows & Matrices

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.

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