deletions | additions       diff --git a/untitled.tex b/untitled.tex  index f6a47a6..d3442ee 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
As $(\frac{W·V}{||W||^2})$ is a constant, so $V$ is a multiple of $W$.  2.   -2.19. Suppose C is an n by n matrix with orthonormal columns. Use Theorem 2.2  to show that  $$||CX|| \leq \sqrt{n} ||X||$$ 
...
Since norms are nonnegative, we can conclude that $||CX|| = ||X||$.  3.   2.44. Use the Cauchy-Schwarz inequality  $$|A · B| \leq||A||||B||$$  to prove:  (a) the function $f (X) = C · X $is uniformly continuous,  (b) the function $g(X, Y) = X · Y$ is continuous.  Answer  (a) In case 1, If $C=0$ then $f(X)=0$ for all $X$,so $|f(X)−f(Y)|=0<ε$ for all $ε,X,Y$.  In case 2, where $C=(c1,c2,...,cn)\neq(0,0,...,0)$.By the definition of f and properties of the dot product, $|f(X)− f(Y)|=|C·Y−C·Y|=|C·(X−Y)|$.  By the Cauchy-Schwartz inequality we get  $|f(X)− f(Y)|=|C·(X−Y)|\leq||C||||X−Y||$.  Let $ε>0$ and take δ=$\sqrt {ε}{||C||}$ . If ||C||  then  ||X−Y||<δ= ǫ ||C||  |f(X)− f(Y)|≤||C||||X−Y||<ǫ  for all X and Y in Rn.