Linear Algebra

196 Chapter Three. Maps Between SpacesRepresenting this vector from the domain with respect to the domain’s basis⎛⎜2⎞ ⎛⎟ ⎜1⎞⎟Rep B ( ⎝2⎠) = ⎝2⎠1 1gives this matrix-vector product.⎛⎜2⎞()⎟ 1 0 −1Rep D ( π( ⎝1⎠) ) =−1 1 11BB,D⎛⎜1⎞⎟⎝2⎠1B( )0=2Expanding this representation into a linear combination of vectors from D( ) ( ) ( )2 1 20 · + 2 · =1 1 2checks that the map’s action is indeed reflected in the operation of the matrix.(We will sometimes compress these three displayed equations into one⎛⎜2⎞ ⎛ ⎞1 ( ) ( )⎟ ⎜ ⎟ h 0 2⎝2⎠ = ⎝2⎠↦−→ =H 2 21 1Din the course of a calculation.)BWe now have two ways to compute the effect of projection, the straightforwardformula that drops each three-tall vector’s third component to make atwo-tall vector, and the above formula that uses representations and matrixvectormultiplication. Compared to the first way, the second way might seemcomplicated. However, it has advantages. The next example shows that thisnew scheme simplifies the formula for some maps.1.9 Example To represent a rotation map t θ : R 2 → R 2 that turns all vectors inthe plane counterclockwise through an angle θD⃗ut π/6−→t π/6 (⃗u)we start by fixing bases. Using E 2 both as a domain basis and as a codomainbasis is natural, Now, we find the image under the map of each vector in thedomain’s basis.(10)(t↦−→θ) ( )cos θ 0sin θ 1(t↦−→θ)− sin θcos θThen we represent these images with respect to the codomain’s basis. Becausethis basis is E 2 , vectors represent themselves. Adjoining the representations