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3.III. Computing Linear Maps. 3.III.1. Representing Linear Maps with Matrices 3.III.2. Any Matrix Represents a Linear Map. 3.III.1. Representing Linear Maps with Matrices. A linear map is determined by its action on a basis. Example 1.1 :. h : R 2 → R 3 by. Let. →. →. →. →.
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3.III. Computing Linear Maps 3.III.1. Representing Linear Maps with Matrices 3.III.2. Any Matrix Represents a Linear Map
3.III.1. Representing Linear Maps with Matrices A linear map is determined by its action on a basis. Example 1.1: h: R2 → R3 by Let → → → →
Given E.g. Matrix notation:
Definition 1.2: Matrix Representation Let V and W be vector spaces of dimensions n and m with bases Band D. The matrix representation of linear map h: V → W w.r.t. Band D is an mn matrix where Example 1.3: h: R3 → P1by Let
Definition 1.5: Matrix-Vector Product The matrix-vector productof a mn matrix and a n1 vector is Theorem 1.4: Matrix Representation Let H = ( hi j) be the matrix rep of linear map h: V n → W mw.r.t. bases Band D. Then where Proof: Straightforward (see Hefferon, p.198 )
Example 1.6: (Ex1.3) h: R3 → P1by Task: Calculate where h sends or
Example 1.7: Let π: R3 → R2 be the projection onto the xy-plane. And → Illustrating Theorem 1.4 using → →
Example 1.8: Rotation Let tθ : R2 → R2 be the rotation by angle θ in the xy-plane. → E.g. Example 1.10: Matrix-vector product as column sum
Exercise 3.III.1. 1. Assume that h: R2 → R3 is determined by this action. Using the standard bases, find (a) the matrix representing this map; (b) a general formula for h(v). 2. Let d/dx: P3 →P3 be the derivative transformation. (a) Represent d/dx with respect to B, Bwhere B = 1, x, x2, x3 . (b) Represent d/dx with respect to B, Dwhere D= 1, 2x, 3x2, 4x3 .
3.III.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. Proof by construction: Given an mn matrixH = ( hi j), one can construct a homomorphism h: V n → W m by v h(v) with h(vB)D = H· vB where B and D are any bases for V and W, resp. vB is an n1 column vector representing vV w.r.t. B.
Example 2.2: Which map the matrix represents depends on which bases are used. Let Then h1: R2 → R2 as represented by H w.r.t. B1 and D1 gives While h2: R2 → R2 as represented by H w.r.t. B2 and D2 gives Convention: An mnmatrix 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).
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 → Rm. ∴ 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 mn matrix H. Then h is onto rank H = m h is 1-1 rank H = n
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 R2 to P1 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.
Exercise 3.III.2. 1. Decide if each vector lies in the range of the map from R3 to R2 represented with respect to the standard bases by the matrix. (a) (b) 2. Describe geometrically the action on R2 of the map represented with respect to the standard bases E2 , E2 by this matrix. Do the same for these:
