chapter 5

Question 1

An inner product or scalar product on a vector space V over R is a map that associates to each pair of vectors u,v∈V a real number〈u,v〉such that the following rules are satisfied for all vectors u,v,w∈V andλ∈R:

Answer 1

(i) 〈u,v〉=〈v,u〉(symmetry),
(ii) 〈v,v〉≥0and〈v,v〉=0 if and only if v=0 (positivedefiniteness),
(iii) 〈u+v,w〉=〈u,w〉+〈v,w〉(linearity1),
(iv) 〈λu,v〉=λ〈u,v〉(linearity2).

Question 2

A vector space endowed with an inner product is called an

Answer 2

inner product space

Question 3

(Cauchy-Schwarz inequality) Let V be an inner product space and let u, v ∈ V . Then

Answer 3

|<u>| ≤ ||u||||v||, and the equality happens if and only if u and v are collinear.</u>

Question 4

Let V be an inner product space and let u, v ∈ V . (i) The vectors u,v are said to be orthogonal,if

Answer 4

if〈u,v〉=0.

(ii) If the vector u is orthogonal to each vector x in a set S⊂V, then we say that u is orthogonal to S.

Question 5

(Pythagoras)Two vectors u and v are orthogonal if and only if

Answer 5

||u+v||^2=||u||^2+ ||v||^2.

Question 6

If a vector v is orthogonal to u1,…,uk,

Answer 6

then v is orthogonal to every vector of span(u1,…,uk).

Question 7

An orthogonal family of vectors that does not contain 0 is

Answer 7

linearly independent

Question 8

Let V be a finite dimensional vector space, and let W be a vector subspace. For a vector v∈V,an orthogonal projection of v on W is a vector πW(v)∈W such that:

Answer 8

〈v−πW(v),w〉=0 for every w∈W.

If W is the line generated by a vector u, we simply denote by πu(v) the orthogonal projection of v on W .

Question 9

Let V be a finite dimensional vector space, and let W be a vector subspace. For a vector v ∈ V , the orthogonal projection πW (v ) satisfies:

Answer 9

||v − πW (v )|| < ||v − w || for every w ∈ W \ {πW (v )}.

Hence, the orthogonal projection is unique and minimises the distance between v and a vector ofW.