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Sections 1.8 and 1.9: Linear Transformations Definitions: 1. Transformation: A transformation from Rn to Rm is a rule that assigns a vector in Rn to a vector in Rm . 2. Domain: The domain of a transformation is the input set. For example, if T : Rn → Rm then dom(T ) = Rn . 3. Image: If x ∈ dom(T ) then T (x) is the image of x under the action of T . 4. Codomain: The codomain of a transformation is the set containing the images of the domain under T . For example, if T : Rn → Rm then codom(T ) = Rm . 5. Range: The range of a transformation is the set of all images ran(T ) = {T (x) : x ∈ dom(T )} 6. Matrix Transformation: Let T : Rn → Rm where x 7→ Ax. If T is defined in this way then it is a matrix transformation. 7. Linear Transformation: A mapping, T , is linear if: (a) T (u + v) = T (u) + T (v) for all u and v in dom(T ), and (b) T (cu) = cT (u) for all c ∈ R and for all u ∈ R 8. Linear Transformation (alternate definition): A mapping, T , is linear if T (cu + dv) = cT (u) + dT (v) for all u, v ∈ dom(T ) and for all c, d ∈ R. Note: This definition is the same as the one above. If c = d = 1 we get (a), and if d = 0 we get (b). Theorems: Theorem 1. Every matrix transformation is linear. Proof. Let T (x) = Ax where A is an m × n matrix. Let u, v ∈ dom(T ) = Rn and let c, d ∈ R. To prove that T is linear we must show that T (cu + dv) = cT (u) + dT (v). Indeed, T (cu + dv) = A(cu + dv) (definition of T ) = A(cu) + A(dv) (distributing matrix multiplication) = cAu + dAv (commuting scalar multiplication) = cT (u) + dT (v) (definition of T ) Therefore T is linear. 1 Theorem 2. Let T : Rn → Rm be a linear transformation. There there exists a unique matrix A such that T (x) = Ax for all x ∈ Rn . In fact, A is the m × n matrix whose j th column is the vector T (ej ): h A = T (e1 ) T (e2 ) ··· i T (en ) Question: Let T : Rn → Rm . What do you need to know in order to generate A? Proof. (insert proof here) 2 Special Transformations: Let T : Rn → Rm be a linear transformation. 1. T is one-to-one if each b ∈ Rm is the image of at most one x ∈ Rn 2. T is onto if each b ∈ Rm is the image of at least one x ∈ Rn Questions: 1. If T : R3 → R4 is it possible for T to be one-to-one? Explain. 2. If T : R3 → R4 is it possible for T to be onto? Explain. 3. If T : R4 → R3 is it possible for T to be one-to-one? Explain. 4. If T : R4 → R3 is it possible for T to be onto? Explain. 3 Theorem 3. Let T : Rn → Rm be a linear transformation. T is one-to-one IF AND ONLY IF the equation T (x) = 0 Theorem 4. Let T : Rn → Rm be a linear transformation such that T (x) = Ax. (a) T is onto IF AND ONLY IF the columns of A span (b) T is one-to-one IF AND ONLY IF the columns of A are Example: Let T : R4 → R3 such that x1 + 2x2 T (x) = 2x2 + x4 + x3 . x2 − x4 1. Find the standard matrix A such that T (x) = Ax. 2. What are the domain, codomain, and range of T ? 3. Is T one-to-one? 4. Is T onto? 4