Proper format for writing basis in linear algebra

In summary, when asked to find a basis for a vector subspace, you can provide one possible option by writing three vectors as a set, or you can write the whole span of those vectors. You can also write the basis using parameters or in its simplest form. When asked to present all possible bases, you can write the basis as a set of three vectors or a normalized basis. The vector space V spanned by the basis can be denoted as a linear combination of the vectors in the basis.
  • #1
Dell
590
0
if i am asked to find a basis for a vector subspace, am i meant to jus write one possible option, for example

basis: (8,0,0)(0,6,0)(0,0,2)

or am i supposed to write the whole span

basis: sp{(8,0,0)(0,6,0)(0,0,2)}

or am i meant to write it with parameters

basis: a(8,0,0) + b(0,6,0) + c(0,0,2)

or am i meant to write it in its simplist form

basis: (1,0,0)(0,1,0)(0,0,1)


and how about when i am asked to present all possible basises
 
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  • #2
The basis is a set which consists of three vectors, so I would write:
B = { (8,0,0), (0,6,0), (0,0,2) }
just like you would write set of three numbers like S = {1, 6, 12}.
If you want to give a normalized basis (or that is asked of you), you can write
[itex]\hat B[/itex] = { (1, 0, 0), (0, 1, 0), (0, 0, 1) }

The vector space V which is spanned by that basis is denoted by
V = sp(an) B = sp( { (8,0,0), (0,6,0), (0,0,2) } )
which means that you can write an arbitrary vector v in V as a linear combination
a(8,0,0) + b(0,6,0) + c(0,0,2)

Is that clear?
 

Related to Proper format for writing basis in linear algebra

What is the proper format for writing a basis in linear algebra?

The proper format for writing a basis in linear algebra is to list out the basis vectors in a column vector or as a linear combination of basis vectors. For example, if the basis vectors are v1 = [1,0] and v2 = [0,1], the basis can be written as B = {[1,0], [0,1]} or B = {v1, v2}.

How do I determine if a set of vectors is a basis in linear algebra?

To determine if a set of vectors is a basis in linear algebra, you can use two methods. The first method is to check if the vectors are linearly independent, meaning that no vector in the set can be written as a linear combination of the other vectors. The second method is to check if the vectors span the entire vector space. If both of these conditions are met, then the set of vectors is a basis.

What is the importance of using a proper format for writing a basis in linear algebra?

The proper format for writing a basis in linear algebra is important because it allows for clear and concise communication of mathematical concepts. It also helps to avoid confusion and errors when performing calculations and proofs involving basis vectors.

Can a basis in linear algebra be written in a different format?

Yes, a basis in linear algebra can be written in different formats. Some common alternative formats include using row vectors instead of column vectors, or using a matrix with the basis vectors as its columns. However, it is important to follow a consistent format when working with basis vectors to avoid confusion.

How do I write a basis for a subspace in linear algebra?

To write a basis for a subspace in linear algebra, you can use the same methods as determining a basis for a vector space. Check if the vectors in the subspace are linearly independent and if they span the subspace. If they do, then the set of vectors can be written as a basis for the subspace.

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