Notes on Inner Products and The Spectral Theorem - 1 Inner...

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1 Inner Products 1.1 Introduction In MAT223, we discussed the dot product of vectors in R n . x 1 x 2 . . . x n · y 1 y 2 . . . y n = x 1 y 1 + x 2 y 2 + · · · + x n y n R . Example 1.1. In R 3 , 1 0 - 2 · 3 5 1 = 3 · 1 + 0 · 5 - 2 · 1 = 1 . Proposition 1.2: Properties of the Dot Product Let x, y, z R n be vectors, and let a, b R be scalars. Then 1. ( ax + by ) · z = a ( x · z ) + b ( y · z ) 2. x · y = y · x 3. x · x 0 and x · x = 0 if and only if x = 0 . Recall that using the dot product, we can define the length of a vector in R n , and we can measure angles between vectors. Definition 1.3. 1. If v R n , we define the length (or norm) k v k to be k v k = v · v, 2. If x, y R n are two non-zero vectors, then the angle θ between x and y is defined 1
to be θ = cos - 1 | x · y | k x kk y k , 0 θ π 2 . Example 1.4. 1. Find the length of the vector v = 1 2 3 R 3 . 2. Find the angle between the vectors x = 1 0 and y = " 1 2 1 2 # . Solution. 1. k v k = v · v = 1 · 1 + 2 · 2 + 3 · 3 = 14 . 2. θ = cos - 1 x · y k x kk y k = cos - 1 1 / 2 1 ! = π 4 . In MAT224, we work with abstract vector spaces and arbitrary fields. Question : If V is any vector space over a field F , can we define some structure on V similar to the dot product on R n that allows us to think about lengths and angles? Answer : If F = R or C then yes, but otherwise no. Remark : Such structures are called inner products . Definition 1.5. Let V be a vector space over R . An inner product on V is a mapping , ·i : V × V R (ie if x, y V we get h x, y i ∈ R ) such that 2
1. h ax + by, z i = a h x, z i + b h y, z i for all x, y, z V and a, b F , 2. h x, y i = h y, x i for all x, y V 3. h x, x i ≥ 0 for all x V , and h x, x i = 0 if and only if x = 0 . We call V with , ·i a real inner product space . Definition 1.6. Let V be a vector space over C . An inner product on V is a mapping , ·i : V × V C (ie if x, y V then h x, y i ∈ C ) satisfying 1. h ax + by, z i = a h x, z i + b h y, z i for all x, y, z V and a, b C (same as before) 2. h x, y i = h y, x i for all x, y V 3. h x, x i ≥ 0 for all x V , and h x, x i = 0 if and only if x = 0 . We call V with , ·i a complex inner product space . Question : Why is ( 2 ) different in this second definition? Answer : If x V then h x, x i = h x, x i implies that h x, x i ∈ R . This is required in order for ( 3 ) to make sense. Example 1.7. 1. The vector space V = R n with the inner product h x, y i = x · y . This shows that the dot product on R n is an example of an inner product, called the standard inner product on R n . So, R n with the dot product is a real inner product space. 2. Let V = C n and suppose that z = z 1 . . . z n , w = w 1 . . . w n . Define h z, w i := z 1 ¯ w 1 + z 2 ¯ w 2 + · · · + z n ¯ w n . (1) This is an inner product and is called the standard inner product on C n . So, C n with , ·i is a complex inner product space.

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