Section 6.1: Inner product, length, and orthogonality

Definition: Let 𝑢,𝑣∈ℝ𝑛. The inner product of 𝑢 and 𝑣 is the number ___, also called a dot product and denoted 𝑢⋅𝑣.

𝑢^𝑇𝑣

properties of the inner product

a. 𝑢⋅ 𝑣 = ?

b. (𝑢+𝑣)⋅𝑤 = ?

c. (𝑐𝑢)⋅𝑣 =𝑐(?) = 𝑢(?)

d. 𝑢⋅ 𝑢≥ 0, and 𝑢⋅𝑢=0 if and only if 𝑢=?

a) 𝑣⋅𝑢

b) (𝑢⋅𝑤)+(𝑣⋅𝑤)

c) (𝑢⋅𝑣), (𝑐𝑣)

d) 0

a vector with length 1 is called a?

unit vector

we can find a unit vector in the same direction by normalizing it using what formula?

v / ||v||

Definition: The distance between 𝑢,𝑣∈ℝ𝑛 is the length of ?

𝑢−𝑣, 𝑑𝑖𝑠𝑡(𝑢,𝑣)=||𝑢−𝑣||

Formula: (geometric definition of the dot product) Let 𝜃 be the angle between the vectors 𝑢 and 𝑣, then 𝑢⋅𝑣 =?

𝑢⋅𝑣=||𝑢||||𝑣||cos𝜃

Definition: 𝑢,𝑣∈ℝ𝑛 are orthogonal (or perpendicular) if ?

𝑢⋅𝑣=0.

the zero vector is ____ to every vector.

orthogonal

Theorem: (The Pythagorean theorem) Vectors 𝑢 and 𝑣 are orthogonal if and only if ||𝑢+𝑣||²= ?

||𝑢+𝑣||² = ||𝑢||²+||𝑣||²

Definition: Let 𝑊 be a subspace of ℝ𝑛. If the vector 𝑧 is orthogonal to every vector in 𝑊, then z is _____. The set of all vectors orthogonal to 𝑊 is called the ______ to 𝑾 and is denoted 𝑊⊥ (read “𝑊 perp”).

orthogonal to every vector in W, orthogonal complement

Properties:

1. A vector 𝑥 is in 𝑊⊥ if and only if ?

2. 𝑊⊥ is ?

1) 𝑥 is orthogonal to every vector in a set that spans 𝑊 (in particular a basis)

2) a subspace of ℝ𝑛

Theorem: Let 𝐴 be an 𝑚×𝑛 matrix. Then (𝑅𝑜𝑤 𝐴)⊥=? , (𝐶𝑜𝑙 𝐴)⊥= ?

1)𝑁𝑢𝑙 𝐴

2) 𝑁𝑢𝑙 𝐴^𝑇

T/F: v dot v = ||v||²

false

T/F) u⋅v − v⋅u=0.

true

(T/F) If the distance from u to v equals the distance from u to −v, then u and v are orthogonal.

True

(T/F) If ∥u∥²+∥v∥²=∥u+v∥², then u and v are orthogonal.

true

T/F) If vectors v1, …, vp, span a subspace W and if x is orthogonal to each vj for j=1,…,p, then x is in W⊥.

true

(T/F) For any scalar c, ∥cv∥=c∥v∥.

true

T/F) For any scalar c, u⋅(cv) = c(u⋅v).

true

T/F) For a square matrix A, vectors in Col A are orthogonal to vectors in Nul A.

false

T/F) For an m×n matrix A, vectors in the null space of A are orthogonal to vectors in the row space of A.

true