I still don't quite get the idea of subspaces, span, and range

[kesun]

Alright. As stated in the title above.

So, a subspace is a set of vectors that satisfies:
1) It contained the zero vector;
2) It's closed under addition and subtraction.

By "closed", it means that when I add another vector in R² or multiply by a scalar k on A(x)=m, it will end up with A(x)+A(y)=m or k(A(x))=m.

Is it correct so far?

Now, I know span is like mapping, but I don't have a precise definition nor explanation for it...Is it like mapping a spanning set to a set of column spaces?

Range is all possible result of a linear transformation of a matrix, ie. L(x)=b, where b is a spanned vectors..is it?

I need some clear definitions and examples to sort this out...Thanks! :D

 

[Ben Niehoff]

It's easiest to imagine what a subspace is in three dimensions, and then generalize to higher dimensions. In 3-dimensional Euclidean space, a subspace is any 3-, 2-, 1-, or 0-dimensional "slice" of space that passes through the origin. That is, all planes or lines that pass through the origin are subspaces, as is the origin itself. One example of a subspace is the x-y plane; another example is the line going through the point (2, 1, 5). Planes and lines of all other orientations are also included, provided that they pass through the origin.

The span of a given set of vectors is precisely the subspace defined by those vectors. For example, the vectors (1,0,0) and (0,1,0) span the x-y plane. The vectors (1,1,0) and (1,-1,0) also span the x-y plane. The vector (2,1,5) spans the 1-dimension subspace given in the last paragraph.

Basically, the span of a set of vectors a_m is the subspace of all vectors which can be written as linear combinations of a_m. The vectors a_m then constitute a basis for this subspace.

 

[kesun]

The word definitions are clear, but I find it hard to actually put them to use in problems.

For example, if a question asks to find the matrix of a linear mapping L:R² -> R² that has a nullspace of span({(2, 1)}) and range span({(2, 1)}), what's the method of determine this matrix? I understand that I am supposed to put the givens into use, but the straightforward of only givens of spans kind of stuck me of how to manipulate them to give me what I want. For example, how is this range denoted in normal column space notation (or something that's easier to understand)?

 

[yyat]

For any vectors $$v_1,...,v_m$$ in $$\mathbb{R}^n$$, the span of these vectors is the range of the linear map $$\mathbb{R}^m\to\mathbb{R}^n$$ given by the matrix with these vectors as columns.

For example, if a question asks to find the matrix of a linear mapping L:R2 -> R2 that has a nullspace of span({(2, 1)}) and range span({(2, 1)}), what's the method of determine this matrix?

A linear map is not uniquely determined by its image and range, so you will need to make some arbitrary choices. For example, we could choose the first column to be (2,1), i.e. (1,0) is mapped to (2,1) by the corresponding linear map. (Any other vector not in the span of (2,1) would also work.)
Since we do not want the image to be larger than the span of (2,1), the other basis vector, (0,1) must be mapped to a vector of the form (2a,a). For (2,1) to be in the kernel of the resulting map, is equivalent to
2*2 + 2a*1=0
1*2 + a*1=0
So one must take a=-2, hence a solution is the matrix with columns (2,1), (-4,-2).

 

[kesun]

Alright!

So for example, if I was asked to fine the matrix of a linear mapping L:R² -> R³ whose nullspace is Sp({(1,1)}) and range is Sp({(1,2,3)}), then the matrix turns out to be:

[1 -1]
[2 -2]
[3 -3]

right?

 

[yyat]

Alright!

So for example, if I was asked to fine the matrix of a linear mapping L:R² -> R³ whose nullspace is Sp({(1,1)}) and range is Sp({(1,2,3)}), then the matrix turns out to be:

[1 -1]
[2 -2]
[3 -3]

right?

Yes.

 

[kesun]

Hmm...I just found this weird problem:

Let n and m be distinct non-zero vectors in R³, and let b be an arbitrary vector in R². Is W={b$$\in$$R²|(n [dot] x, m [dot] x)=b for some x in R³} a subspace of R²? R³?

First of all, it concerns both R² and R², and I am not sure what's the approach for something like this.. :S

 

[yyat]

Do the vectors of W have 2 or 3 components? This should tell you where W might be a subspace.
Then you will need to check the axioms of a subspace: closure under addition and scalar multiplication, are they satisified?

 

[HallsofIvy]

Alright. As stated in the title above.

So, a subspace is a set of vectors that satisfies:
1) It contained the zero vector;
2) It's closed under addition and subtraction.

By "closed", it means that when I add another vector in R² or multiply by a scalar k on A(x)=m, it will end up with A(x)+A(y)=m or k(A(x))=m.

Is it correct so far?

No. It must also contain the product of a real number (or whatever the base field is) and a vector. For example the set of all integers is closed under addition and subtraction but is not a subspace of R¹. Also, since v- v= 0, you don't need "contains 0" but you do need "contains at least one vector".

Now, I know span is like mapping, but I don't have a precise definition nor explanation for it...Is it like mapping a spanning set to a set of column spaces?

I wouldn't think of the span as being anything like a "mapping"! The span of a set of vectors is the set of all possible linear combinations of the vectors.

Range is all possible result of a linear transformation of a matrix, ie. L(x)=b, where b is a spanned vectors..is it?

.
I don't know what you mean by "b is a spanned vector"- you have no spanning set. I wouldn't say "linear transformation of a matrix". A matrix is a way of representing a linear transformation, given a specific basis. Certainly, the range of a linear transformation is a the set of all possible "results": {b| Ax= b for some x}. It is, in fact, exactly the same as the general "range" of a function.

I need some clear definitions and examples to sort this out...Thanks! :D