m235notes13-page2 - Proof. By definition, the set B spans...

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Unformatted text preview: Proof. By definition, the set B spans V . This means exactly that every element in V can be written as a linear combination of the elements of B , that is, we can write an arbitrary element w 2 V as m X w= ai v i . 1 We have to show that this expression for w is unique. Assume that we have another expression for w, say m X w= bi vi . 1 Subtracting we obtain w w= m X ( ai bi ) vi = 0. 1 This is a linear relation among the elements of B . The only such relation is the trivial one. Hence ai = bi . This says that there is only one expression for w as a linear combination of elements of B . Coordinates Definition 1. Let A = {v1 , v2 , · · · vn } be a basis of a subspace of Rm . Let v 2 V , so we can uniquely write n X v= ai v i . 1 We say that the coordinates of v are a1 , a2 , · · · , an . We write 01 a1 B a2 C v=B C . @ a3 A ·A Example 2. Let 01 01 ✓◆ 0 0 1 @1A , e3 = @0A}. E = {e1 = e= 02 0 1 01 x 3 @y A we have This is a basis of R and for an arbitrary element v = z 01 x @y A . v = xe1 + xe2 + xe3 , and so v = zE This says that the coordinates of v wrt the basis E are x, y, x. 2 ...
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