Please show middle proof step by step Let T R2 rightarrow R2

Please show middle proof step by step.

Let T: R^2 rightarrow R^2 be an orthogonal linear transformation. Show that T is either a rotation or a reflection.

Solution

Let T:R²R² be an orthogonal linear transformation . Let its standard matrix be A =

m

n

p

q

Since A is orthogonal, we must have AA^T = A^TA = I₂ where I₂ is the 2x2 identity matrix. Therefore, we have m²+n² =1…(1), p²+q² = 1…(2) and mp+nq = 0…(3). Now, let m = cos and p = sin. Then, in view of the equations (1), (2) and (3), we must have either n = -p and q = m or, n = p and q = -m.

Further, if n = -p and q = m, then A =

cos

-sin

sin

cos

This shows that A represents a counterclockwise rotation by .

Also, if n = p and q = -m, then A =

cos

sin

sin

-cos

In this case the matrix A represents a reflection across a line with slope (1/2).

Thus,if T:R²R² be an orthogonal linear transformation then T is either a rotation or a reflection.

m	n
p	q