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- The dot product
Definition 21.1. The dot product of two vectors ~u and ~v in Rn^ is the sum ~u · ~v = u 1 v 1 + u 2 v 2 + · · · + unvn.
Example 21.2. The dot product of ~u = (1, 1) and ~v = (2, −1) is
~u · ~v = 1 · 2 + 1 · −1 = 1.
The dot product of ~u = (1, 2 , 3) and ~v = (2, − 1 , 1) is
~u · ~v = 1 · 2 + 2 · −1 + 3 · 1 = 3. Note that when we compute the product of two matrices A and B in essence we are computing an array of dot products. In particular the dot product can be identified with the matrix product ~uT^ · ~v.
Definition 21.3. The length of a vector ~v ∈ Rn^ is the square root of the dot product of ~v with itself:
‖~v‖ =
~v.~v =
v 12 + v^22 + · · · + v n^2.
Note that ‖(x, y)‖ =
x^2 + y^2 and ‖(x, y, z)‖ =
x^2 + y^2 + z^2
the usual formula for the length, using Pythagoras.
Example 21.4. What is the length of the vector ~v = (1, − 2 , 2)?
~v · ~v = 1^2 + 2^2 + 2^2 = 9.
So the length is 3. Note that the vector
uˆ =
~v = (1/ 3 , − 2 / 3 , 2 /3)
is a vector of unit length with the same direction as ~v.
Definition 21.5. Let p and q be two points in Rn. The distance between P and Q is the length of the vector ~v = q − p. Let p = (1, 1 , 1) and q = (2, − 1 , 3). Then ~v = (2, − 1 , 3) − (1, 1 , 1) = (1, − 2 , 1).
So the distance between p and q is 3, the length of ~v.
Definition 21.6. We say two vectors are orthogonal if ~u · ~v = 0.
The standard basis vectors ~e 1 , ~e 2 ,... , ~en of Rn^ are orthogonal. Example 21.7. Are the vectors ~u = (1, 1 , −2) and ~v = (2, 0 , 1) orthog- onal?
~u · ~v = (1, 1 , −2) · (2, 0 , 1) = 2 + 0 − 2 = 0, so that ~u and ~v are orthogonal. Definition-Theorem 21.8. Let W ⊂ Rn^ be a linear subspace. The orthogonal complement of W is W ⊥^ = { v ∈ Rn^ | v · w = 0 }, the set of all vectors which are orthogonal to every vector in W. Then W ⊥^ is a linear subspace of Rn. For example, suppose we start with a plane H in R^3 through the origin. Then there is a line L in R^3 through the origin which is the orthogonal complement of H: L = H⊥. The line L is spanned by a vector which is orthogonal to every vector in H. Note that the relation between L and H is reciprocal, H is the orthogonal complement of L: H = L⊥. Theorem 21.9. Let A be an m × n matrix. The orthogonal complement of the row space of A is the null space of A and the orthogonal complement of the column space of A is the null space of AT^ : (Row A)⊥^ = Nul A and (Col A)⊥^ = Nul AT^. Proof. The rows of A correspond to equations. If a row is given by the vector ~a = (a 1 , a 2 ,... , an) then the corresponding equation is a 1 x 1 + a 2 x 2 + · · · + anxn = 0.
~x is in the null space if and only if it satisfies every equation. But ~x satisfies the equation a 1 x 1 + a 2 x 2 + · · · + anxn = 0 if and only if the dot product ~a · ~x = 0. Thus ~x is in the null space if and only if it is in the orthogonal complement of the row space.
