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Hi Alex,
answers to your questions below:
-you are right that collinearity is another way of saying linear
dependency. see Fox p. 528 or Simon and Blume Ch. 11. When a vector is
linearly dependent, it can be expressed as a linear function of another
vector or set of vectors.
In the 2-D case, collinearity is quite intuitive. Two vectors are
linearly independent if they do not overlap or radiate out in opposite
directions from the origin, which only happens when they
are multiples of one another. Think about a vector x= (3,2) extending from
the origin to x1=3,x2=2 (Figure 5.B in Fox) Any linear function of this
vector (where our new vector y= a +bx) would be colinear. For example, in
this diagram, y=1.5x which yields y=(4.5,3). This vector falls in the
same line, i.e., is "collinear" with our original vector x.
As we move into more dimensions (i.e., our vectors are of length k>2),
this same logic applies. A vector is linearly dependent if we can be
calculated as a linear combination of a set of basis vectors. Let's
consider the 3-D case from the section handout. Notice that y can be
defined as a linear function of x1, x2, and x3. This means that y falls
within the 3-D space generated by x1, x2, and x3 (I am assuming that
x1,x2, and x3 are linearly independent, i.e., none of these is a linear
function of the other).
We can speak of linear independence when a vector cannot be expressed as
a linear combination of a set of other vectors. In other words, it does
not fall within the subspace generated by our other vectors. If two
vectors together define a plane, if our third vector juts off into 3-D
space, we cannot define it in terms of the vectors that define the plane.
It is linearly independent.
If you find Fox a bit dense, you might take a quick look at the first few
pages of Simon and Blume chapter 11.
On Thu, 11 Mar 2004, Alex Liebman wrote:
> hey alison,
>
> quick question, or, actually, just want to confirm that this true:
>
> 1) if two vectors are not collinear, then they are linearly independent.
>
> 2) if three vectors are not collinear (that is, x is not collinear with y,
> nor y with z, nor z with x), they are still linearly dependent.
>
> what i'm really trying to say is that collinearity and linear independence
> are related, but actually different. is that right?
>
> thanks,
> alex
>
>
