richard r
Poster

Post Number: 14

Posted on Tuesday, 19 April, 2005 - 03:11 pm:

Hello,

Could someone please help me with this topic. What is meant by a vector subspace ? What are linearly independent vectors, and what is a basis ? Also, what does it mean to say vectors "span" a subspace ?

Some simple explanations would be really appreciated.

Thanks

Richard

Tristan Marshall
Regular poster

Post Number: 61

Posted on Tuesday, 19 April, 2005 - 03:44 pm:

Do you know what a vector space is? How would you define a vector?

If you know what a vector space is, then a vector subspace is a subset (of a vector space) that is also a vector space.

A simple example would be if we take our vector space to be $R^{3}$ (the space consisting of three ordered co-ordinates). The set of elements with the last co-ordinate equal to zero forms a subspace.

A set $V$ of vectors spans a subspace $W$ if any element $W$ can be written as a sum of elements in $V$ .

Two vectors $v_{1}$ , $v_{2}$ are said to be linearly independent if and only if the only solution to $λ_{1} v_{1} + λ_{2} v_{2} = 0$ is $λ_{1} = λ_{2} = 0$ .

For thevector space $R^{2}$ (i.e. 2-D Cartesian co-ordinate vectors) thistranslates as 'the vectors don't point in the same direction'.

Similarly, a set of $n$ vectors $v_{1}$ , ..., $v_{n}$ are said to be linearly independent if and only if the only solution to $λ_{1} v_{1} + \dots + λ_{n} v_{n} = 0$ is $λ_{1} = \dots = λ_{n} = 0$ .

A basis is a linearly independent spanning set. It can be proved that all bases contain the same number of elements, which wecall the dimension.

James
Veteran poster

Post Number: 792

Posted on Tuesday, 19 April, 2005 - 04:24 pm:

I know I'm just repeating whats been said, lol, but I've written it all out now.

A vector subspace is just a vector space itself, but all the vectors are also in a larger vector space. Eg. a line is a subspace of a plane, a plane is a subspace of R³ .

A set of vectors, {u₁ ,u₂ ...u_n } are linearly independent vectors if the only solution to
a₁ u₁ + a₂ u₂ + ... a_n u_n = 0
is where all the a_i = 0

Eg, in R² , the vectors (1,0) and (0,1) are linearly independent, but (3,0) and (7,0) are not, as
7(3,0) - 3(7,0) = (0,0) = 0

A set of vectors {u₁ ,u₂ ...u_n } which span the space have the property that, for any vector, v in the space,
v = a₁ u₁ + a₂ u₂ + ... a_n u_n
has at least 1 solution. So every single vector can be made from some combination of the u_i .

A basis is a spanning set which is linearly independent.

In R³ , {(0,0,1), (0,1,0), (1,0,0), (3,7,4)} spans the space (but isnt a basis), as
(x,y,z) = x(1,0,0) + y(0,1,0) + z(0,0,1) + 0(3,7,4)
But you can see that we have a redundant vector in the spanning set, we dont actually need all 4. We can get rid of any 1 of them and still have a spanning set(I've just shown how you can get rid of (3,7,4) because its most obvious, but you can get rid of any of the other 3 instead). A basis is a spanning set where we have gotten rid of all the redundant vectors. It means that for any vector, v,
v = a₁ u₁ + a₂ u₂ + ... a_n u_n
has a unique solution.