Question

Let b-,-1,1). Define T:RR by the mapping: V3 T(x)-(x b)b (a) Show that T is a linear transformation by verifying the two linear transformation axioms. b) Determine the standard matrix representation for 1 (c) Give a geometrical interpretation of T

Answer 1

(a). Let x and y be 2 arbitrary vectors in R³ and let k be an arbitrary real scalar.

Then T(x+y) = ((x+y).b)b = (x.b+y.b)b = (x.b)b+(y.b)b = T(x)+T(y). Hence, T preserves vector addition.

Also, T(kx) = (kx.b)b = k(x.b)b = kT(x). Hence, T preserves scalar multiplication.

Therefore, T is a linear transformation.

(b). We have T(e₁) = (e₁.b)b = (1/√3)(1/√3,-1/√3,1/√3) = (1/3,-1/3,1/3), T(e₂) = (e₂.b)b = -(1/√3)(1/√3,-1/√3,1/√3) = (-1/3,1/3,-1/3) and T(e₃) = (e₃.b)b = (1/√3)(1/√3,-1/√3,1/√3) = (1/3,-1/3,1/3). Hence the standard matrix of T is A(say) =

1/3	-1/3	1/3
-1/3	1/3	-1/3
1/3	-1/3	1/3

(c ). Since both e₁ and e₃ are mapped to the same vector (1/3,-1/3,1/3), hence the image of T is a plane in R³, being the span of 2 linearly independent vectors in R³ .