Is a Vector e=(0,1) a Basis in R^2?

Thread starter sibiryk
Start date Sep 26, 2005
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Basis Vector

In summary, a basis in R^2 is a set of linearly independent vectors that span the entire vector space. E=(0,1) is a basis in R^2 because it is a single vector that is both linearly independent and spans the entire vector space. To determine if a vector is a basis in R^2, you need to check if it is linearly independent and if it spans the entire vector space. A vector with more than two dimensions cannot be a basis in R^2 because R^2 is a two-dimensional vector space. Not all vectors in R^2 are considered a basis, as a basis must be a set of linearly independent vectors that span the entire vector space. A vector with all

Sep 26, 2005

sibiryk

I am not sure if a vector e=(0,1) is a basis in R^2.

Can it give an expansion to vector u:

u=a(0,1)

u1=0
u2=a ?

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Sep 27, 2005

sibiryk

I got it. It a was a stupid question.

Sep 27, 2005

HallsofIvy

Science Advisor

Homework Helper

R² is "2 dimensional" and so requires 2 vectors for a basis.

1. Is e=(0,1) a Basis in R^2?

Yes, e=(0,1) is a Basis in R^2. A Basis is a set of linearly independent vectors that span the entire vector space. Since e=(0,1) is a single vector, it is by definition linearly independent and since it spans the entire R^2 vector space, it is a basis.

2. How do you determine if a vector is a basis in R^2?

To determine if a vector is a basis in R^2, you need to check if it is linearly independent and if it spans the entire vector space. This can be done by checking if the vector can be multiplied by any scalar and still remain in the vector space, and if it can be combined with other vectors in the vector space to form any other vector in the space.

3. Can a vector with more than two dimensions be a basis in R^2?

No, a vector with more than two dimensions cannot be a basis in R^2. R^2 is a two-dimensional vector space, so a basis in R^2 must also have two dimensions. A vector with more dimensions cannot span the entire R^2 vector space.

4. Are all vectors in R^2 considered a basis?

No, not all vectors in R^2 are considered a basis. A basis must be a set of linearly independent vectors that span the entire vector space. So, while every vector in R^2 can be part of a basis, not all vectors can be a basis on their own.

5. Can a vector with all zeroes be a basis in R^2?

No, a vector with all zeroes cannot be a basis in R^2. To be a basis, a vector must be linearly independent and span the entire vector space. A vector with all zeroes cannot be linearly independent and cannot span the entire R^2 vector space.