Orthonormal Sets A set S = {v 1 , v 2 , …, v n } is an orthonormal set if the dot product between every vector in S is zero AND the norm(magnitude) of each vector in S is one. If S is not an orthonormal set, then we can make it an orthonormal set by first using a tedious procedure called the Gram-Schmidt Algorithm. This algorithm turns a set of vectors into and orthogonal set. After the use of the algorithm, we make the set orthonormal by dividing each vector in the orthogonal set by its corresponding magnitude. Gram-Schmidt Algorithm: Given S = {v 1 , v 2 , …, v n }, find an orthogonal set S’ = {u 1 , u 2 , …, u n }. 1. Let u 1 = v 1 . 2. Compute u 2 = v 2 – proj u 1 v 2 3. Compute u 3 = v 3 – proj u 1 v 3 – proj u 2 v 3 : : : Step n. Compute u n = v n – proj u 1 v n – proj u 2 v n - … - proj u n – 1 v n If you forgot how to compute a projection here is how you can remember. B B B A A B 2 ⋅ = proj *Remember, this is read as “the projection of vector A onto vector B”. The big letter appears once in the formula where the small letter appears three times. Example: Turn the following set into an orthonormal set. { (1, -1), (2, 0)} 1. Use the Gram-Schmidt Algorithm to turn the set into an orthogonal set. ) 1 , 1 ( 1 − = u ) 1 , 1 ( ) 1 , 1 ( ) 0 , 2 ( ) 1 , 1 ( 2 2 ) 0 , 2 ( ) 1 , 1 ( ) 1 ( ) 1 ( ) 1 , 1 ( ) 0 , 2 ( ) 0 , 2 ( 2 2 2 2 2 1 = − − = − − = − − + − ⋅ − = − = v v u u proj 2. Divide each vector found in step one by its corresponding norm(magnitude). This will then be the orthonomoral set. 2 ) 1 ( ) 1 ( 2 2 1 = − + = u 2 ) 1 ( ) 1 ( 2 2 2 = + = u Therefore, the orthonormal set is given by − 2 1 , 2 1 , 2 1 , 2 1
A set S = {v1, v2, , vn} is an orthonormal set if the dot
product between every vector in S is zero AND the norm(magnitude)
of each vector in S is one. If S is not an orthonormal set, then we
can make it an orthonormal set by first using a tedious procedure
called the Gram-Schmidt Algorithm. This algorithm turns a set of
vectors into and orthogonal set. After the use of the algorithm, we
make the set orthonormal by dividing each vector in the orthogonal
set by its corresponding magnitude. Gram-Schmidt Algorithm: Given S
= {v1, v2, , vn}, find an orthogonal set S = {u1, u2, , un}. 1. Let
u1 = v 1. 2. Compute u 2 = v 2 proj u 1 v 2 3. Compute u 3 = v 3
proj u 1 v 3 proj u 2 v 3 : : : Step n. Compute u n = v n proj u 1
v n proj u 2 v n - - proj u n 1v n If you forgot how to compute a
projection here is how you can remember.
B
B
BAA
B 2
=proj
*Remember, this is read as the projection of vector A onto
vector B. The big letter appears once in the formula where the
small letter appears three times. Example: Turn the following set
into an orthonormal set. { (1, -1), (2, 0)} 1. Use the Gram-Schmidt
Algorithm to turn the set into an orthogonal set.
)1,1(1 =u
)1,1()1,1()0,2()1,1(2
2)0,2()1,1(
)1()1(
)1,1()0,2()0,2(
22222 1===
+
== vvu
uproj
2. Divide each vector found in step one by its corresponding
norm(magnitude). This will then be the orthonomoral set.
2)1()1( 221 =+=u 2)1()1(22
2 =+=u
Therefore, the orthonormal set is given by
2
1,2
1,
2
1,2
1
You can check to see if this is correct by computing the
corresponding dot products and taking the norms of each vector.
Remember if all dot products equal zero and the norms of all the
vectors equal one, then you know you have the right orthonormal
set. Check:
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