Page 424 - 35Linear Algebra

424 Movie Scripts

Finally we calculate
0
t 3 = m 3 − (m 0 1 m 3 )m − (m 0 2 m 3 )m 0 2
1
√ 2
0
0
0
0
1
2
= m 3 − r m − r m = m 3 + 2m − √ m ,
1
3
3
2
1
2
3
q
0
0
0
3
again noting m 0 m = km k = 1, and let m = t 3 where kt 3 k = r = 2 2 . Thus
2 2 2 3 kt 3 k 3 3
we get our final M = QR decomposition as
√
 1 1 1  √ 
√ √ − √ 2 0 − 2
2 3 2 √
q  0
3 .
Q =  0 √ 1 2  , 3 √ 2 


3 3 R =  q
− √ 1 1 − √ 1 0 0 2 2 3
2 3 6
Overview
This video depicts the ideas of a subspace sum, a direct sum and an orthogonal
3
complement in R . Firstly, lets start with the subspace sum. Remember that
even if U and V are subspaces, their union U ∪ V is usually not a subspace.
However, the span of their union certainly is and is called the subspace sum
U + V = span(U ∪ V ) .
You need to be aware that this is a sum of vector spaces (not vectors). A
3
picture of this is a pair of planes in R :
3
Here U + V = R .
Next lets consider a direct sum. This is just the subspace sum for the
case when U ∩ V = {0}. For that we can keep the plane U but must replace V by
a line:

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