linear-guest

14
Orthonormal Bases and Complements
You may have noticed that we have only rarely used the dot product. That
is because many of the results we have obtained do not require a preferred
notion of lengths of vectors. Once a dot or inner product is available, lengths
of and angles between vectors can be measured–very powerful machinery and
results are available in this case.
14.1 Properties of the Standard Basis
The standard notion of the length of a vector x = (x 1 , x 2 , . . . , x n ) ∈ R n is
||x|| = √ x x = √ (x 1 ) 2 + (x 2 ) 2 + · · · (x n ) 2 .
The canonical/standard basis in R n
⎛ ⎞ ⎛ ⎞
⎛ ⎞
1
0
0
e 1 = ⎜ ⎟
⎝
0. ⎠ , e 2 = ⎜ ⎟
⎝
1. ⎠ , . . . , e n = ⎜ ⎟
⎝
0. ⎠ ,
0
0
1
has many useful properties with respect to the dot product and lengths.
• Each of the standard basis vectors has unit length;
‖e i ‖ = √ √
e i e i = e T i e i = 1 .
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