How Does a Vector with 3 Components Span a Plane in Linear Algebra?

[tomizzo]

I am not here to ask a question directly out of the book, but rather I want someone to explain to me something about spans in linear algebra.

I realize that spans basically are just the area/region in which vectors can occupy based on different scalars. However, I don't understand how a vector with 3 components can create a "plane" rather than 3 dimensional space. I realize that two vectors can lie on the same plane, but shouldn't the span allow to vectors to occupy spaces above and below the plane as well?

Thanks,

/** I apologize for placing this thread in the wrong place/

 

[Sankaku]

You have started with the right idea. When you start with a vector v and add a new one w, it comes down to whether w adds any new 'information'. If w is collinear (lies on the same line) with v then the space spanned by v and w is still the same as the space (line) spanned by v.

You probably already knew that. However, just extend the idea to planes. If v and u span a plane, when does adding a new vector w cause the span to remain the same, or to change to become a 3-space?

 

[HallsofIvy]

The span of two vectors is the set of all linear combinations of the two vectors- the span of u and v is the set of all vectors au+ bv where a and b are numbers. If, for example, u= <1, 0, 0> and v= <0, 1, 0>, the span is all vectors of the form a<1, 0, 0>+ b<0, 1, 0>= <a, b, 0>. Note that no matter what a and b are, the third component is always 0. I don't know where you got the idea "shouldn't the span allow to vectors to occupy spaces above and below the plane as well?" No, it shouldn't!

 

[Deveno]

I am not here to ask a question directly out of the book, but rather I want someone to explain to me something about spans in linear algebra.

I realize that spans basically are just the area/region in which vectors can occupy based on different scalars. However, I don't understand how a vector with 3 components can create a "plane" rather than 3 dimensional space. I realize that two vectors can lie on the same plane, but shouldn't the span allow to vectors to occupy spaces above and below the plane as well?

Thanks,

/** I apologize for placing this thread in the wrong place/

perhaps i can make this clearer to you with an easy example.

we will start with R³, which we will imagine is like the 3-dimensional space we perceive around us. it has become customary to reference a point (or "vector") in this space, by specifying 3 coordinates (which are each real numbers), the x-coordinate, the y-coordinate and the z-coordinate.

what this means is that we need 3 pieces of information to locate a point in 3-dimensional space (often called "left/right", "back/forth" and "up/down", although this, too, is a somewhat arbitrary convention). such a point is written like so: (x,y,z), as a 3-tuple (triple) of real numbers.

now, let's consider the following span-set:

span({(1,0,0),(0,1,0)}), the span of all linear combinations of the points (vectors): (1,0,0) and (0,1,0), which you might recognize as unit vectors lying on the x-axis, and y-axis, respectively. a typical element in this span set looks like this:

a(1,0,0) + b(0,1,0) = (a,0,0) + (0,b,0) = (a,b,0).

note that the "third coordinate" is always going to be 0, so even though the vectors in the span set all lie in 3-dimensional space, we only need to specify TWO numbers, to identify an element of this span-set (namely, the numbers a and b).

what is this span-set? it is the part of 3-space that is commonly called the xy-plane, a 2-dimensional subset of R³. a "lower dimensional" (dimension here refers not just to our intuitive sense, but also the strict mathematical sense of number of basis elements) vector space, residing in a higher dimensional one, is called a subspace. so here, we have that the xy-plane forms a 2-dimensional subspace of R³. in fact, it behaves like a copy of R² embedded in R³.

perhaps this example will help you understand that the dimension of the "enveloping space" (in this case, the number of coordinates) has little bearing on the dimension of a subspace residing in it (except, of course, that a subspace cannot be of larger dimension that the space it lives in).

now there is no reason to restrict "generating vectors" to lying on the x-,y- or z-axis. in fact, the choice of these axes is somewhat arbitrary. other coordinate systems to describe 3-space are certainly possible (such as spherical, or cylindrical coordinates). the basic rules (in low dimensions) are:

one linearly independent (non-redundant) vector determines a line (1-dimensional space)
two linearly independent vectors determine a plane (2-dimensional space)

higher-dimensional examples are possible to describe mathematically, but rather hard to visualize or draw explicitly.