Just need help with parts a,b, and e. We say an n times n matrix A is orthogonal

Question

Just need help with parts a,b, and e.

We say an n times n matrix A is orthogonal if AT A = In. Prove that the column vectors a1an of an orthogonal matrix A are unit vectors that ore orthogonal to one another, i,e., Fill in the missing columns in the following matrices to make them orthogonal: Show that any 2 times 2 orthogonal matrix A must be of the form for some real number theta, ( Hint: Use part a, rather than the original definition,) Show that if A is an orthogonal 2 times 2 matrix, them mu A: R2 rightarrow R2 is either a rotaion or the composition of a rotation and a reflection Prove that the row vectors A1An of an orthogonal matrix A are unit vectors that are orthogonal to one another, (Hint: Corollary 3,3)

Explanation / Answer

a.

A^T.A = I

=> i,j the element of A^T.A = 1, if i=j and 0 otherwise

=> (ith row of A^T).(jth column of A) = 1 if i=j and 0 otherwise

=> dot product of ith and jth columns of A = 1, if i=j and 0 otherwise (since ith row of A^T is same as ith column of A)

=> column vectors of A are unit vectors (since dot product of ith column with itself is 1) and are orthogonal to each other (since dot product of i and jth columns, i distinct from j is 0).

(b)

Using a, we need to fill missing column so that it is a unit vector and orthogonal to all other columns.

for the first matrix fill the column as (1/2; (root3)/2) or -(1/2; (root3)/2) since we need (a;b) such that

(root3/2).a +(-1/2).b = 0 and a^2+b^2 = 1.

for the second matrix, orthogonality with first column gives us that first entry of the required column is 0, and similarly orthogonality with second column gives second entry also as 0.

It being a unit vector gives the last remaining entry as 1 or -1.

Third matrix we need to find a,b,c such that

(1/3).a+(2/3).b+(2/3).c = 0

(2/3).a+(-2/3).b+(1/3).c = 0

a^2+b^2+c^2 = 1

first two eqns give

a+2b+2c = 0

2a-2b+c = 0

<=> 3a+3c = 0, a+2b+2c =0 (first eqn+second eqn, second eqn)

<=> a=-c, c=-2b

<=> (a.b.c) = (2,1,-2).k for some real k.

Now a^2+b^2+c^2 = 1 gives k as 1/3 or -1/3

thus required column is (2/3,1/3,-2/3) or -(2/3,1/3,-2/3)

(e)

I suppose corollary 3.3 is if A is orthogonal A^T is orthogonal.

Using this result A^T is orthogonal,

so column vectors of A^T are unit vectors orthogonal to each other

In other words row vectors of A are unit vectors orthogonal to each other.

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