Linear Algebra Step by Step notes Chapter 4

Posted on August 12, 2015 by Sumant Sumant

Chapter 4

For a general vector space, the inner product is denoted by <u,v> rather than u.v

Definition: An inner product on a real vector space V is an operation which assigns to each pair of vectors u and v which satisfies the following axioms for all vectors u,v and w in V and all scalars k.

  • <u,v> = <v,u>
  • <u+v,w> = <u,w>+<v,w>
  • <ku,v> =k<u,v>
  • <u,u> \ge 0

Proposition 4.17

Let \{v_1,v_2,v_3,...,v_n \} be an orthonormal set of vectors in an inner product space V of dimension n. Let u be a vector in V then

u= <u,v_1>v_1+<u,v_2>v_2+<u,v_3>v_3...+<u,v_n>v_n

Definition 4.18

A square matrix Q=(v_1,v_2,v_3,...,v_n), whose columns v_1,v_2,v_3,...,v_n are orthonormal (perpendicular unit) vectors, is called an orthogonal matrix

Proposition 4.19

Let Q=(v_1 v_2 v_3 ... v_n) be a square matrix. Then Q is an orthogonal matrix \iff Q^{T}Q=I

Proposition 4.20

Q is an orthogonal matrix \iff Q^{-1}=Q^T

Proposition 4.21

Let Q be an n by n matrix and u and v be vectors in R^n. Then Q is an orthogonal matrix \iff Q^{u} \cdot Q^{w} = u \cdot w

Proposition 4.22

Let Q be an n by n matrix and u be a vector in R^n. We have Q is an orthogonal matrix \iff ||Qu||=||u||

Proposition 4.23

A triangular matrix is an n by n matrix where al entries to one side of the leading diagonal are zero.

Definition 4.24 A diagonal matrix is an n by n matrix where all entries to both sides of the leading diagonal are zero.

Proposition 4.25

$A=QR$

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