Basics

Here are a few ideas you need to know about.  Maybe you've seen them before.

The row way of computing A x dots each row of A with x.
E.g., (2 3) (4) = (<2, 3> · <4, 3>) = (8 + 9) = (17)    1 2 3 <1, 2> · <4, 3> 4 + 6 10.

The column way of computing A x forms a linear combination of the columns of A using weights from x .
E.g. (2 3) (4) = 4 (2) + 3 (3) = ( 8 ) + ( 9 ) = ( 17 ) 1 2 3 1 2 4 6 10.

An orthonormal basis for ^n is a set of n orthogonal unit vectors.  
E.g. {(3/5) , (-4/5) } 4/5 3/5 is an orthonormal basis for ^2.

Orthonormal coordinates
If v is a vector in ^2, and B = {u_1, u_2} is a basis for ^2 you can write v = c_1u_1 + c_2u_2 for some constants c_i.  The numbers c_i are the u_i-coordinates of v .  When B = {u_1, u_2} is an orthonormal basis the u_i-coordinates for v have a simple formula:  c_i = u_i · v .  

Example.  Write v = (5) 5 as a linear combination of u_1 = (3/5) 4/5 and u_2 = (-4/5) 3/5.  
Since B = {u_1, u_2} is an orthonormal basis you compute the u_i coordinates using dot products:  (5) = (u_1 · v) u_1 + (u_2 · v) u_2 = 7u_1 + (-1) u_2 = 7 (3/5) + (-1) (-4/5) 5 4/5 3/5.

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Copyright © 2007 Todd Will
Created by Mathematica  (April 15, 2007)