Geometrical Interpretation of V in Span S

[Nano-Passion]

What does it mean geometrically to say that V lies in span S, in other words that V is in a linear combination of S?

 

[A Dhingra]

Let me just try and answer this (because i am studying the same thing and was trying to interpret it geometrically). Let us consider two objects (rather vectors, mind well not the ones that have mag. and direction) u1 and v1[itex]\in[/itex] S, such that they are linearly independent. Example (1,1),(0,1). Or simply consider u1=(0,1) and v1=(1,0), ([itex]\hat{i}[/itex], [itex]\hat{j}[/itex])and using these two ordered pair you can form all the other ordered pairs and extend it to the whole V.
It is like saying, a(1,0)+b(0,1)=(x,y), i.e. their linear combination gives the whole region occupied by ordered pair. that is same as saying these vectors u1 and v1 cover or span the whole vector space V2(2-d vector space in this case). this same idea can be extended to n-dimensional vector space.
It is also possible that with a span S(u1, u2, u3) you span a plane of 3-d vector space depending upon the vectors u1, u2, u3 (this case comes when one of the three is dependent on the other two).
hope this helps.

 

[Vorde]

Geometrically, let's restrict ourselves to R^3. And let us say we are talking about the span of two vectors ##\vec{v}## and ##\vec{w}##. If you remember from earlier math classes, for any two vectors there exists one and only one plane such that both vectors lie in the plane.

This plane that they both lie in can be considered as the sum of all the points that are a linear combination of S, a.k.a. the sum of all points that can be made by adding a scalar multiple of ##\vec{v}## to a scalar multiple of ##\vec{w}##.

Therefore, if you had a third vector ##\vec{x}## that was in span(##\vec{v}##,##\vec{w}##), then you are saying that ##\vec{x}## is in that plane defined by ##\vec{w}## and ##\vec{v}##.

Edit: I brushed over some things, like the fact that if ##\vec{v}## and ##\vec{w}## are scalar multiples of each other (or if one is the zero vector) then this doesn't quite work, but hopefully you get the idea.

 

[Nano-Passion]

Geometrically, let's restrict ourselves to R^3. And let us say we are talking about the span of two vectors ##\vec{v}## and ##\vec{w}##. If you remember from earlier math classes, for any two vectors there exists one and only one plane such that both vectors lie in the plane.

This plane that they both lie in can be considered as the sum of all the points that are a linear combination of S, a.k.a. the sum of all points that can be made by adding a scalar multiple of ##\vec{v}## to a scalar multiple of ##\vec{w}##.

Therefore, if you had a third vector ##\vec{x}## that was in span(##\vec{v}##,##\vec{w}##), then you are saying that ##\vec{x}## is in that plane defined by ##\vec{w}## and ##\vec{v}##.

Edit: I brushed over some things, like the fact that if ##\vec{v}## and ##\vec{w}## are scalar multiples of each other (or if one is the zero vector) then this doesn't quite work, but hopefully you get the idea.

Thanks! So essentially V lies in Span S if it also lies in the plane of S.

You gave me an example of two vectors laying in the plane. But what about when there are three vectors in Span S that do not all lie in a common plane?

 

[AlephZero]

Geometrically, you can think of span S as the set of all points that you can reach from the origin, traveling only in directions parallel to the vectors in S.

If S contains two vectors that define a plane, you can get to every point in the plane.

If S contains more than two vectors all in the same plane, that makes no difference - you still can't get to any point that is NOT in the plane.

If S contains three vectors not all in the same plane, you can get to any point in 3-dimensional space.

 

[Nano-Passion]

Geometrically, you can think of span S as the set of all points that you can reach from the origin, traveling only in directions parallel to the vectors in S.

If S contains two vectors that define a plane, you can get to every point in the plane.

If S contains more than two vectors all in the same plane, that makes no difference - you still can't get to any point that is NOT in the plane.

If S contains three vectors not all in the same plane, you can get to any point in 3-dimensional space.

Thanks.

So literally the geometric interpretation is that V lies in S, in which S itself spans a 3-dimensional place. Now can I politely ask, what is the motivation behind this? What does it apply to?

 

[AlephZero]

Now can I politely ask, what is the motivation behind this? What does it apply to?

It's very hard to give an answer when we don't know how much you know already.

How much linear algebra have you studied? What applications of LA as a whole (not just this topic) do you know about?

 

[Nano-Passion]

It's very hard to give an answer when we don't know how much you know already.

How much linear algebra have you studied? What applications of LA as a whole (not just this topic) do you know about?

Just an introductory linear algebra class. I don't know much about applications of LA first hand, but I do know it is applied in many things that involves a large numbers of computations, things such as digital processing to be specific, or computational neuroscience to be general (I plan to pursue that field or mathematical neuroscience). Mathematically, LA has applications in fields such as topology.