1. 3.II.1. Representing Linear Maps with Matrices 3.II.2. Any Matrix Represents a Linear Map 3.II. Computing Linear Maps

2. 3.II.1. Representing Linear Maps with Matrices A linear map is determined by its action on the bases. Consider h : V n → W m with bases B and D for V and W, resp. Range space of h = Range h = span h( B ) = subspace of W. Rank h = dim[Range h ]  min( n, m ) Definition:Matrix Representation The matrix representation of linear map h: V n → W m w.r.t. B and D is an m  n matrix. where

3. Example: Let h: R 2 → R 3 by Given

4. Let → → → →

5. E.g.

6. Example: h: R 3 → P 1 by Let Task: Calculate where h sends

7. Example 1.7: Let π: R 3 → R 2 be the projection onto the xy-plane. And → Illustrating Theorem 1.4 using→ →

8. Example 1.8:Rotation Let t θ : R 2 → R 2 be the rotation by angle θ in the xy-plane. → E.g. Example 1.10:Matrix-vector product as column sum

9. Exercise 3.II.1. Using the standard bases, find (a) the matrix representing this map; (b) a general formula for h(v). 1. Assume that h: R 2 → R 3 is determined by this action. 2. Let d/dx: P 3 → P 3 be the derivative transformation. (a) Represent d/dx with respect to B, B where B =  1, x, x 2, x 3 . (b) Represent d/dx with respect to B, D where D =  1, 2x, 3x 2, 4x 3 .

10. 3.II.2. Any Matrix Represents a Linear Map Theorem 2.1: Every matrix represents a homomorphism between vector spaces, of appropriate dimensions, with respect to any pair of bases. h : V n → W m with bases B and D s.t. Proof by construction: Let H be an m  n matrix. Then there is an homomorphism

11. Example 2.2: Which map the matrix represents depends on which bases are used. Let Then h 1 : R 2 → R 2 as represented by H w.r.t. B 1 and D 1 gives While h 2 : R 2 → R 2 as represented by H w.r.t. B 2 and D 2 gives   Convention: An m  n matrix with no spaces or bases specified will be assumed to represent h: V n → W m w.r.t. the standard bases. In which case, column space of H = R (h).

12. Theorem 2.3: rank H = rank h Proof: (See Hefferon, p.207.) For each set of bases for h: V n → W m,  Isomorphism: W m → R m. ∴ dim columnSpace = dim rangeSpace Example 2.4: Any map represented by must be of type h: V 3 → W 4 rank H = 2 →dim R (h) = 2 Corollary 2.5: Let h be a linear map represented by an m  n matrix H. Then h is onto  rank H = m  All rows L.I.( dim R = dim W ) h is 1-1  rank H = n  All cols L.I.( dim R = dim V )

13. Corollary 2.6: A square matrix represents nonsingular maps iff it is a nonsingular matrix. A matrix represents an isomorphism iff it is square and nonsingular. Example 2.7: Any map from R 2 to P 1 represented w.r.t. any pair of bases by is nonsingular because rank H = 2. Example 2.8: Any map represented by is singular because H is singular. Definition: A matrix is nonsingular if it is square and the corresponding linear system has a unique solution. A linear map is nonsingular if it is one-to-one.

14. Exercise 3.II.2. 1. Decide if each vector lies in the range of the map from R 3 to R 2 represented with respect to the standard bases by the matrix. (a)(b) 2. Describe geometrically the action on R 2 of the map represented with respect to the standard bases E 2, E 2 by this matrix. Do the same for these: