Parametric description of span

[nolita_day]

I'm having some trouble internalizing the concept of span.

The question:

If u = [1,2,1]; v = [-2,2,4]; and w = [-1,4,5], describe Span{u,v,w}.

The attempt at a solution:

I formed a matrix using column vectors u, v, and w and row-reduced to RREF:

$\begin{bmatrix} 1 & -2 & -1 \\ 2 & 2 & 4 \\ 1 & 4 & 5 \end{bmatrix}$ ~ $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix}$

1) Given the equation Ax = b with solutions x₁, x₂, and x₃, where b is the vector [b₁, b₂, b₃] and b₃ must = 0 in order for the equation to be consistent, does that mean Span{u,v,w} must lie in the plane x₃ = 0 in R³?

2) Since I saw that b₃ = 0 was the only condition for this system to be consistent, I assumed Span{u,v,w} had to be a plane. Is there any way for me to see this parametrically by seeing that there are two free variables? I don't know if that even makes sense to ask...

So far in class we have been able to describe solution sets to matrix equations parametrically, so if the parametric description for the solution set of some equation was in the form x₁v₁ + x₂v₂, I could see that x₁ and x₂ are the free variables (and x₃ is the basic variable). So in this case I could see geometrically that the solution set is a plane. Is there an analogous way to interpret the span?

By the way, we have not talked about vector spaces yet so that probably wouldn't help me in an explanation... thanks so much in advance!

 

[chiro]

Hey nolita_day and welcome to the forums.

The span of a set of vectors is basically a linear combination of vectors that can be used to represent all vectors in that set.

If your matrix reduced to the identity, then it means that all vectors are linearly independent and that that none can be written in terms of linear combinations of the others.

Since the matrix is of full rank the span corresponds to the whole space or that of a linear combination of all the vectors or of any basis of R^3 with three parameters required to represent a vector in your set corresponding to ax + by + cz for a,b,c being the parameters and x,y,z being the vectors in your span.

If you get zero rows as you did a row reduction, this would indicate that some vectors are linearly dependent and the number of parameters to describe the space (i.e. the stuff in the spanning set) would be less than 3 for this case.

 

[Mastersbn]

Span is the space of vectors, in which all vectors in the span will be able to represent as the linear combination of its fundamental vectors. eg: a span of vectors (1,0) and (0,1) is the entire 2 dimensional space. Because any vector in 2_d space can be represented as a linear combination of these vectors. eg: suppose (6,7) is a vector which can be represented as 6*(1,0)+7*(0,1). Linear combination of basis vectors. Generally 2 independent vectors can span a 2 dimensional space, 3 independent vectors can span a 3 dimensional space etc. So n independent basis vectors are needed to form a n dimensional space. Here in your example the set of vectors are (1,0,0), (0,1,0) and (1,1,0), which all are not linearly independent and so cannot span a 3 dimensional space. But here if we take any two vectors they are independent of each other and so these vectors can span a 2 dimensional space or span of these vectors will be a 2 dimensional space or a plane.

 

[homeomorphic]

1) Given the equation Ax = b with solutions x1, x2, and x3, where b is the vector [b1, b2, b3] and b3 must = 0 in order for the equation to be consistent, does that mean Span{u,v,w} must lie in the plane x3 = 0 in R3?

Yes, but I don't know if that's very enlightening. The concept is really that if you think of u, v, and w as arrows, they all lie flat in the same plane, so they span that plane. It's clear that this is true for the column vectors of the matrix you ended up with. But it works for the ones you started with because row reduction can be done using operations that don't change the dimension of the span (since you have not talked about vector spaces, it may be hard to understand this very intuitively).

2) Since I saw that b3 = 0 was the only condition for this system to be consistent, I assumed Span{u,v,w} had to be a plane. Is there any way for me to see this parametrically by seeing that there are two free variables? I don't know if that even makes sense to ask...

Pretty much.

So far in class we have been able to describe solution sets to matrix equations parametrically, so if the parametric description for the solution set of some equation was in the form x1v1 + x2v2, I could see that x1 and x2 are the free variables (and x3 is the basic variable). So in this case I could see geometrically that the solution set is a plane. Is there an analogous way to interpret the span?

That is pretty much the concept of span, except you seem to be missing the concept of linear dependence. To see what the span of two vectors is, you just take all linear combinations of them, which is a different way of saying what you said about the two parameters. A set of vectors is linearly dependent if you can throw some of them out without affecting the span. An example is your problem there. You can throw one of them out and they still span the same plane. So, actually, if you take linear combinations of 3 vectors that lie in the same plane, you have three parameters, but since they lie in the same plane, you aren't going to be able to get anything outside the plane, so even thought there are three parameters, the span is 2-d.

A set of vectors is linearly independent if throwing any of them out decreases the dimension of the span. So, in that case if you had 3 vectors, they would span a 3-d space with the three parameters.

 

[blue_raver22]

Dear student,

Thank you for reaching out for help with understanding the concept of span. Parametric description of span refers to the set of all possible linear combinations of a given set of vectors. In your case, the set of vectors is u = [1,2,1], v = [-2,2,4], and w = [-1,4,5].

To understand the span of these vectors, we can think of them as arrows or directions in three-dimensional space. When we take a linear combination of these vectors, we are essentially adding or subtracting a multiple of each vector to create a new vector in the same direction. For example, if we take the linear combination 2u + 3v - w, we get a new vector that is a combination of these three original vectors.

The span of these vectors, Span{u,v,w}, is therefore the set of all possible vectors that can be created by taking linear combinations of u, v, and w. In other words, it represents a plane in three-dimensional space that is formed by all the possible directions or combinations of these three vectors.

To answer your questions:

1) Yes, since the third row of the RREF matrix is [0 0 0], it means that the third component of the solution vector must be 0 in order for the system to be consistent. This implies that the span of u, v, and w lies in the plane x3 = 0 in R3.

2) Yes, the parametric description of the span can be seen by looking at the two free variables in the RREF matrix, x1 and x2. These variables represent the coefficients of u and v respectively, while the basic variable x3 is the coefficient of w. So, we can interpret the span as a plane in three-dimensional space that is formed by the linear combinations of u and v, with w as the basic vector.

I hope this explanation helps you to better understand the concept of span. Keep up the good work in your studies of linear algebra!

Best regards,