# Vectors - Span #RR^2#?

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Any help here is much appreciated

Any help here is much appreciated

##### 1 Answer

A requirement for any two vectors to span

For convenience we normally use a natural basis for vectors based on a standard cartesian coordinate system. Thus we normally use standard vectors

# bb(B_1) = { bb(ul hat i), bb(ul hat j) } = { ((1),(0)), ((0),(1)) } #

Using this basis

# ((3),(2)) = 3bb(ul hat i) + 2bb(ul hat j) = 3((1),(0)) + 2((0),(1)) #

However we can readily show that the vectors:

# bb(ul u) = ((1),(1)) \ \ # and#bb(ul v)= ((-3),(2))#

are also linearly independent, and as such can also be used as a basis:

# bb(B_2) = { bb(ul u), bb(ul v)} = { ((1),(1)), ((-3),(2)) } #

And now to represent the coordinate

# ((3),(2)) = lamda bb(ul u) + mu bb(ul v) #

# \ \ \ \ \ \ \ \ = lamda ((1),(1)) + mu ((-3),(2)) #

# \ \ \ \ \ \ \ \ = ((1,-3),(1,2)) ((lamda),(mu)) #

And solving this system, yields the solution:

# lamda= 12/5 \ \ # , and#mu=-1/5#

Thus we can write the coordinate using this spanning basis

# ((3),(2)) = 12/5 ((1),(1)) -1/5 ((-3),(2)) #