A function f is called onetoone iff fx1 fx2 implies x1 x2

A function f is called one-to-one iff f(x_1) = f(x_2) implies x_1 = x_2. Given a unit vector u epsilon R^n, let U = span(u). Consider a linear transformation L R^n rightarrow R^n defined by L(x) = 2Proj_u x - x. Prove that L is one-to-one.

Solution

Given that L(x) = 2 Proj_U X-X

Since span(u) = U

any vector y in U can be represented as linear combination of u.

i.e. y =c u for a scalar c

Hence L(x1) = 2 Proj x1 on u - x1

= 2 (x1.cu)x1 -x1

= 2c (x1.u)x1-x1

Similarly L(X2) = 2c (x2.u)x2-x2

If L(x1) = L(x2)

then 2c (x1.u)x1-x1 = 2c (x2.u)x2-x2

x1-x2 = 2c (x1.u)x1-2c (x2.u)x2

This is possible only if x1 = x2

Hence L is one to one.