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I really really need help understanding "span" in linear algebra. How do you determine if a vector is the span of a matrix? How do you show that R^2 or R^3 = span of a matrix?
prob) determine if the vector b is in the span of the columns of the matrix A.
A =
[1 2]
[3 4]
b =
[5]
[6]
prob) show that R^2 =
span
([1] , [1])
([1] , [-1])
Any help would be nice (Thinking)
problem a)
can you write the vector b as a linear combination of the two column vectors? if you can, then the vector is in the span.
why is this? well consider the field R^2, with the set v = {(1,0), (0,1)}. Geometrically, these vectors 'span' the entire x-y plane, as you can write any vector in that plane in terms of scalar multiples of those two vectors; i.e. any 2d vector in the x-y plane is in span(v). Does this make sense?
You may or may not have done bases yet, but these vectors (1,0) and (0,1) form a basis (or a coordinate system, for want of another term) for R^2. So if you think about it, if you have a set of two non-parallel vectors in R^2, they will span the entire x-y plane, because you can go , say, 'x' units in the direction of one vector, and 'y' units in the direction of the other vector.
problem b)
hopefully this is made clearer now from the explanation above. Try finding out whether an arbitrary vector (x,y) can be expressed as a linear combination of (1,1) and (1,-1), for any real x and y.
Alternatively, it is easily seen that since the two vectors are not parallel (i.e. one is not a scalar multiple of the other) the two vectors form a linearly independent set in R^2, and any linearly independent set of 'n' vectors in R^n will span that space.
