deletions | additions       diff --git a/untitled.tex b/untitled.tex  index b6ca6a1..d0bfbb7 100644  --- a/untitled.tex  +++ b/untitled.tex 
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$$0 \leq ||V−(U·V)U||^2 = ||V||2 −(U·V)^2$$.  Therefore if $U$ is a unit vector and equality holds, then $V = (U · V)U$. Show that equality occurs in the Cauchy Schwarz inequality for two arbitrary vectors $V$ and $W$ only if one of the vectors is a multiple (perhaps zero) of the other vector.  Answer:  In the first case, when $W=0$, the $W$ is a multiple of $V$;  In the second case, when $W$ is nonzero, then consider the unit vector $U = \frac{W}{||W||}$. Then, by the result in the question, it follows that $V = (U · V)U$. Therefore: 
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$$||CX|| = ||X||$$  for such a matrix.    Answer:   (1) First we calculate $C$. Let $C_j$ denote the $j$th column of C. Since C  has orthonormal columns, each $C_j$ has norm 1. 
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(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 $ε,X,Y$. 
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(c) Set $H = h e_j$ in the $i-th $component of the numerator to show that the partial deriva-  tive $f_{i,x_j} (A)$ exists and is equal to the $(i, j)$ entry of $C$.  Answer:  (a) $F$ from $R_n$ to $R_m$ is differentiable at $A$. By the definition of differentiability there is a linear function $L_A(H)$  such that