16 CHAPTER 1. REVIEW OF SOME LINEAR ALGEBRA

By induction, u j =w j and so this reduces to wk+1/ |wk+1|=wk+1. ■This lemma immediately implies the following lemma.

Lemma 1.5.3 Let V be a subspace of dimension p and let {w1, · · · ,wr} be an or-thonormal set of vectors in V . Then this orthonormal set of vectors may be extended toan orthonormal basis for V, {

w1, · · · ,wr,yr+1, · · · ,yp}

Proof: First extend the given linearly independent set {w1, · · · ,wr} to a basis for Vand then apply the Gram Schmidt theorem to the resulting basis. Since {w1, · · · ,wr} isorthonormal it follows from Lemma 1.5.2 the result is of the desired form, an orthonormalbasis extending {w1, · · · ,wr}. ■

Here is another lemma about preserving distance.

Lemma 1.5.4 Suppose R is an m×n matrix with m≥ n and R preserves distances. ThenR∗R = I. Also, if R takes an orthonormal basis to an orthonormal set, then R must preservedistances.

Proof: Since R preserves distances, |Rx|= |x| for every x. Therefore from the axiomsof the dot product,

|x|2 + |y|2 +(x,y)+(y,x) = |x+y|2 = (R(x+y) ,R(x+y))

= (Rx,Rx)+(Ry,Ry)+(Rx,Ry)+(Ry,Rx)

= |x|2 + |y|2 +(R∗Rx,y)+(y,R∗Rx)

and so for all x,y,(R∗Rx−x,y)+(y,R∗Rx−x) = 0

Hence for all x,y, Re(R∗Rx−x,y) = 0. Now for a x,y given, choose α ∈ C such that

α (R∗Rx−x,y) = |(R∗Rx−x,y)|

Then0 = Re(R∗Rx−x,αy) = Reα (R∗Rx−x,y) = |(R∗Rx−x,y)|

Thus |(R∗Rx−x,y)| = 0 for all x,y because the given x,y were arbitrary. Let y =R∗Rx−x to conclude that for all x,

R∗Rx−x= 0

which says R∗R = I since x is arbitrary.Consider the last claim. Let R : Fn→ Fm such that {u1, · · · ,un} is an orthonormal basis

for Fn and {Ru1, · · · ,Run} is also an orthormal set, then∣∣∣∣∣R(

∑i

xiui

)∣∣∣∣∣2

=

∣∣∣∣∣∑ixiRui

∣∣∣∣∣2

= ∑i|xi|2 =

∣∣∣∣∣∑ixiui

∣∣∣∣∣2

■

With this preparation, here is the big theorem about the right polar factorization.