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Linear Algebra Spans Question

  1. Jan 19, 2012 #1
    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/
     
  2. jcsd
  3. Jan 19, 2012 #2
    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?
     
  4. Jan 19, 2012 #3

    HallsofIvy

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    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!
     
  5. Jan 19, 2012 #4

    Deveno

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    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.
     
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