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TheStudent111

Questions on Linear Algebra

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Been studying linear algebra and I have run into some questions I would like some clarification on:

 

Are vector spaces composed of basis vectors? Where the y axis is just j hat and x axis is i hat? Figure below

 

Are linear transformations just the manipulation of basis vectors which result in the movement of the resultant vector? So the phrase "space transform" is essentially the transformation of each basis vector (which results in the change of the vector space). Figure below

Edited by TheStudent111

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Yes, a set of basis vectors spans a linear space. It's a linearly independent set of vectors.

 

A geometric vector is a visualization of a vector. Usually we can draw and visualize its lenght and direction on a paper.

 

Given a linear space S we can define a set of basis vectors that spans S, so that any vector in S can be uniquely identified by a linear combination of the basis vectors and some scalars.

 

v = x * i + y * j

 

Or just the scalars if you already know the basis

 

(x, y)

 

because of linear independence. 

 

In all your sketches all vector sets forms a basis for a linear space because roughly the vectors are orthogonal to each other. Typically i = (1, 0) and j = (0, 1).

 

Yes, linearly mapping a vector v in a space A to B by a transformation T is the same as mapping the basis for A from A to B, not forming another basis for B!

 

T(v) = x * T(i) + y * T(j

Edited by Irlan Robson

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Are vector spaces composed of basis vectors?


No, that sentence is not quite right. Vector spaces are composed of many vectors. A basis is a collection of vectors with specific properties (they are linearly independent and they span the whole vector space).

Are linear transformations just the manipulation of basis vectors which result in the movement of the resultant vector?


The language is imprecise, but you may have the right idea. I'll try to say it more precisely.

A linear transformation between a vector space V and a vector space W is a mapping f such that
* f(u+v) = f(u)+f(v), for all vectors u and v in V; and
* f(s*v) = s * f(v), for all vectors v in V and all scalars s

You can uniquely determine a linear transformation by the images of the vectors in a basis.

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No, that sentence is not quite right. Vector spaces are composed of many vectors. A basis is a collection of vectors with specific properties (they are linearly independent and they span the whole vector space).

Sorry for the wording. So am I wrong to think of basis vectors as the x and y axis (i hat and j hat to be exact) that cover a space? Am I overthinking what a vector space is? Is a vector space just the traditional x and y axis plane that also happens to have i hat and j hat?

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The x and y axes are the directions of the vectors you call i hat and j hat (although that's not a notation I use). I don't know what it means for axes to "cover" a space. There is a notion of "spanning" or "generating" a space, and indeed a basis does span the whole vector space, because being a generator set is part of the definition of basis.

The way I think about it, a vector space is a set of things that can be added and scaled. The traditional plane (thought of as a set of arrows from the origin) is an example of vector space, but there are others. For instance, the set of continuous functions from R to R is another example of vector space (but trying to understand what a basis looks like in this case is very very tricky).

If you really want to understand vectors, I recommend you start here: http://www.3blue1brown.com/essence-of-linear-algebra/ Edited by Álvaro

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Been studying linear algebra and I have run into some questions I would like some clarification on:

Are vector spaces composed of basis vectors? Where the y axis is just j hat and x axis is i hat? Figure below

Are linear transformations just the manipulation of basis vectors which result in the movement of the resultant vector? So the phrase "space transform" is essentially the transformation of each basis vector (which results in the change of the vector space). Figure below

This concept alludes me for quite a while before it clicked one day. Supplement this with the Rodrigues axis rotation (you should be able to derive it yourself as I did several years ago), Petty soon you will be able to write your own rigid body simulator without the need for quaternions.

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The x and y axes are the directions of the vectors you call i hat and j hat (although that's not a notation I use). I don't know what it means for axes to "cover" a space. There is a notion of "spanning" or "generating" a space, and indeed a basis does span the whole vector space, because being a generator set is part of the definition of basis.

The way I think about it, a vector space is a set of things that can be added and scaled. The traditional plane (thought of as a set of arrows from the origin) is an example of vector space, but there are others. For instance, the set of continuous functions from R to R is another example of vector space (but trying to understand what a basis looks like in this case is very very tricky).

If you really want to understand vectors, I recommend you start here: http://www.3blue1brown.com/essence-of-linear-algebra/


+1 This is really cool, thanks for sharing.

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