Linear Algebra Please Help Me Let A be an n Times n orthogonal matrix, i.e. ATA

Question

Linear Algebra Please Help Me

Let A be an n Times n orthogonal matrix, i.e. ATA = I. Show that A preserves angles in RN (i.e. the angle between Ax and Ay is the same as the angle between X and y). [Hint: think about what happens to the inner product of x and y.] Let A be a non-zero n Times n matrix whose column vectors A1, A2, . . . , AN are mutually orthogonal. Let b be a vector in RN. Recall that the equation Ax = b can be written as x1A1+x2A2+...+xnAn=b Use the orthogonality of the columns of A to give a formula for x1,x2, . . . ,xn. [Hint: think about the dot product of each Ai with both sides of the equation above.] Let A be as in part (a). Show that A is non-singular.

Explanation / Answer

a)

Since A^TA=I we can write : <Ax,Ay> = (Ax)^TAy=x^TA^TAy=x^Ty=<x,y>

So the angle are preserved.

b) Write x1A1+..+xnAn=b

Now notice <Ai,Aj>=0 where i!=j

So <b,Ai>=xi<Ai,Ai> ( all the other terms are zero as explained before )

So xi = <b,Ai>/<Ai,Ai>

c) Since AA^T=I then :

det(AA^T)=det(A)det(A^T)=det(A)^2=1.

so det(A)=+-1 , which is never zero, so A is invertible.

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