Chapter 6A Flashcards
Memorize
Dot Product (R) Definition
For x,y ∈ R^(n)
The dot product of x and y is
x . y = x_(1) y_(1) + … +x_(n) y_(n)
where x = (x_(1), x_(2),…, x_(n)) and y = (y_(1), y_(2),…, y_(n))
Inner Product (Real Numbers)
In R^(n), the inner product of
v= (x_(1),… ,x_(n)) and w = (y_(1), …,y_(n))
is < v, w > = x_(1) y_(1) +…+x_(n)y(n)
Inner Product (Complex Numbers)
In C^(n)
the inner product of
v= (x_(1),… ,x_(n)) and w = (y_(1), …,y_(n))
is = x_(1) ȳ_(1) +…+ x_(n) _ȳ(n)
Inner Product Definition + Properties
An inner product on V is a function that takes ordered pairs (u,v) of elements of V to a number <u> ∈ F
Positivity- <u> ≥ 0 for all v ∈ V
Additivity in the first slot -
<u>= <u> + for all u,v,w ∈ V</u></u></u></u>
Definiteness-
= 0 if and only if v=0
Conjugate Symmetry-
<u> = conjugate for all u,v ∈ V</u>
Homogeneity in the first slot-
= λ<u> for all λ ∈ F and for all u,v ∈ V</u></u></u></u></u></u>
Inner Product Space
An inner product space is a vector space V along with an inner product on V
Basic Properties of an inner product
(a) For each fixed u ∈ V, the function that takes v to is a linear map from V to F
(b) <0,u> = 0 for every u ∈ V
(c) <u>=0 for every u ∈ V
(d) <u> = <u> + <u> for all u, v, w ∈ V
(e) <u> = conjugate λ <u> for all λ ∈ F and u,v ∈ V </u></u></u></u></u></u>
Norm
For v ∈ V, the norm of v is
|| v || = squareroot (< v, v >)
Basic Properties of the norm
Suppose v ∈ V
(a) || v || = 0 if and only if v=0
(b) || λv || = |λ| || v || for all λ ∈ F
Orthogonal
Two vector u, v are orthogonal if < u, v > = 0
Orthogonality and 0
(a) 0 is orthogonal to every vector in V
(b) 0 is the only vector in V that is orthogonal to itself
Pythagorean Theorem
Let u and v be orthogonal vectors in V.
Then || u + v ||^(2) = || u ||^(2) + || v ||^(2)
