I've been working on the following problem:
"Let be a finite-dimensional vector space over a field , and let be an ordered basis for . Let be an invertible matrix with entries from . Define for , and set . Prove that is a basis for and hence that is the change of coordinate matrix changing -coordinates into -coordinates."
Can someone point me in the right direction with this? Obviously I need to show that is linearly independent and spans . I know there are n vectors in so it only remains to show that its elements are linearly independent, correct? I can't seem to figure it out though.
in general, if A is a set, and T is a linear transformation (which every matrix is), then T(A) is the set of all T(a) where a is an element of A.
so in this case, . the trick is to show all of these elements are distinct (which you
should be able to do just from the fact that Q is invertible, and hence 1-1). do these n vectors span V? well, they have to, since Q
is invertible, and thus onto (so we can write any vector w in V as Q(v) for some other vector v in V, and we can write v as a linear
combination of the xj, and thus write w = Q(v) as the same linear combination of the Q(xj), since Q is linear).
but, if Q(β) spans V, and has the same cardinality as a basis (this is where the distinctness comes in), it has to be a basis, too,
because it is a minimal spanning set.
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you can also show the linear independence directly:
suppose . then
.
since Q is 1-1, , which means by the
linear independence of the xj that .
this is a recurring theme in linear algebra:
injective/1-1 ~~ linear independence
surjective/onto ~~ spanning
I completely understand how the invertibility of results in a new basis by taking , but I didn't think that was acting as left multiplication on . I feel this is somewhat dumb, but am I really interpreting for incorrectly? I don't see how it amounts to left multiplication by on an element of . This must be where my real confusion lies.