Linear Algebra-Lecture 29 Slides-Mathematics, Slides of Linear Algebra

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MATH 304

Linear Algebra Lecture 29: Orthogonal sets. The Gram-Schmidt process.

Orthogonal sets

Let V be an inner product space with an inner product 〈·, ·〉 and the induced norm ‖ · ‖. Definition. A nonempty set S ⊂ V of nonzero vectors is called an orthogonal set if all vectors in S are mutually orthogonal. That is, 0 ∈/ S and 〈x, y〉 = 0 for any x, y ∈ S, x 6 = y. An orthogonal set S ⊂ V is called orthonormal if ‖x‖ = 1 for any x ∈ S. Remark. Vectors v 1 , v 2 ,... , vk ∈ V form an orthonormal set if and only if

〈vi , vj 〉 =

1 if i = j 0 if i 6 = j

• V = C [−π, π], 〈f , g 〉 =

∫ (^) π

−π

f (x)g (x) dx.

f 1 (x) = sin x, f 2 (x) = sin 2x,... , fn(x) = sin nx,...

〈fm, fn〉 =

∫ (^) π −π

sin(mx) sin(nx) dx

=

∫ (^) π −π

1 2

( cos(mx − nx) − cos(mx + nx)

) dx.

∫ (^) π −π

cos(kx) dx = sin( kkx)

∣∣ ∣

π x=−π^ = 0^ if^ k^ ∈^ Z,^ k^6 = 0. k = 0 =⇒

∫ (^) π −π

cos(kx) dx =

∫ (^) π −π

dx = 2π.

〈fm, fn〉 =^12

∫ (^) π −π

( cos(m − n)x − cos(m + n)x

) dx

=

{ π if m = n 0 if m 6 = n

Thus the set {f 1 , f 2 , f 3 ,... } is orthogonal but not orthonormal.

It is orthonormal with respect to a scaled inner product

〈〈f , g 〉〉 =

π

∫ (^) π

−π

f (x)g (x) dx.

Orthonormal bases

Let v 1 , v 2 ,... , vn be an orthonormal basis for an inner product space V. Theorem Let x = x 1 v 1 + x 2 v 2 + · · · + xnvn and y = y 1 v 1 + y 2 v 2 + · · · + ynvn, where xi , yj ∈ R. Then (i) 〈x, y〉 = x 1 y 1 + x 2 y 2 + · · · + xnyn, (ii) ‖x‖ =

x 12 + x 22 + · · · + x n^2. Proof: (ii) follows from (i) when y = x.

〈x, y〉 =

〈 (^) n ∑ i=

xi vi ,

∑^ n j=

yj vj

〉

∑^ n i=

xi

〈 vi ,

∑^ n j=

yj vj

〉

=

∑^ n i=

∑^ n j=

xi yj 〈vi , vj 〉 =

∑^ n i=

xi yi.

Let v 1 , v 2 ,... , vn be a basis for an inner product space V.

Theorem If the basis v 1 , v 2 ,... , vn is an orthogonal set then for any x ∈ V

x =

〈x, v 1 〉 〈v 1 , v 1 〉 v 1 +

〈x, v 2 〉 〈v 2 , v 2 〉 v 2 + · · · +

〈x, vn〉 〈vn, vn〉 vn.

If v 1 , v 2 ,... , vn is an orthonormal set then

x = 〈x, v 1 〉v 1 + 〈x, v 2 〉v 2 + · · · + 〈x, vn〉vn.

Proof: We have that x = x 1 v 1 + · · · + xnvn. =⇒ 〈x, vi 〉 = 〈x 1 v 1 + · · · + xnvn, vi 〉, 1 ≤ i ≤ n. =⇒ 〈x, vi 〉 = x 1 〈v 1 , vi 〉 + · · · + xn〈vn, vi 〉 =⇒ 〈x, vi 〉 = xi 〈vi , vi 〉.

Orthogonalization

Let V be a vector space with an inner product. Suppose x 1 , x 2 ,... , xn is a basis for V. Let v 1 = x 1 , v 2 = x 2 − 〈x 2 , v 1 〉 〈v 1 , v 1 〉

v 1 ,

v 3 = x 3 −

〈x 3 , v 1 〉 〈v 1 , v 1 〉 v 1 −

〈x 3 , v 2 〉 〈v 2 , v 2 〉 v 2 ,

................................................. vn = xn − 〈xn, v 1 〉 〈v 1 , v 1 〉

v 1 − · · · − 〈xn, vn− 1 〉 〈vn− 1 , vn− 1 〉

vn− 1.

Then v 1 , v 2 ,... , vn is an orthogonal basis for V. The orthogonalization of a basis as described above is called the Gram-Schmidt process.