MA 242 September 28, 2007Two theorems on mapping properties of linear transformationsTheorem. Let T : Rn→ Rmbe a linear transformation. Then T is one-to-one if and onlyif the only solution to T(x) = 0 is the trivial solution.Theorem. Let T : Rn→ Rmbe a linear transformation and let A be its standard matrixrepresentation. Then1. T maps Rnonto Rmif and only if the columns of A span Rm, and2. T is one-to-one if and only if the columns of A are linearly independent.1MA 242 September 28, 2007Matrix algebraMatrices have an unusual algebraic structure associated to them.Matrix addition and scalar multiplication should remind you of vector addition and scalarmultiplication in Rn.Definitions.1. Let A and B be two m × n matrices. Then the sum of A and B is the m × n matrix Cwherecij= aij+ bij.2. Let A be an m × n matrix and r be a real number. Then the scalar multiple rA is thematrix whose ij-entry is raij. In other words, every entry of A is multiplied by r.Theorem 1. Let A, B, and C be m × n matrices and let r and s be real numbers. Then1. A + B = B + A2. (A + B) + C = A + (B + C)3. A + 0 = A4. r(A + B) = rA + rB5. (r + s)A = rA + sA6. r(sA) = (rs)AMatrix multiplication is more subtle than addition or scalar multiplication, and there aretwo ways to define it. One way is algebraic. The other is geometric.2MA 242 September 28, 2007Here is the geometric definition:Let A : Rn→ Rmand B : Rp→ Rnbe linear transformations. What is the matrixrepresentation of the composition A ◦ B? (Note that this composition has B as the firsttransformation.)3MA 242 September 28, 2007Definition. The product C of A and B is the matrixC =AB1AB2. . . ABp.Let’s check the sizes of all of the matrices involved.Example. Let A : R2→ R2be rotation by 45◦and let B : R2→ R2be the matrixtransformation determined by the matrix2 11 1.What is the matrix representation for the transformation A ◦ B?4MA 242 September 28, 2007Row-column dot product definition: The columns of AB are linear combinations ofthe columns of A. In fact, consider the jth column of AB.Row-column rule: (AB)ij= ai1b1j+ ai2b2j+ . . . + ainbnj5MA 242 September 28, 2007Definition. The n × n (square) matrixIn=1 0 . . . . . . 00 1 . . . . . . 0..................0 0 . . . . . . 1with 1’s down the diagonal and 0’s everywhere else is called the n × n identity matrix.Theorem 2. Let A be an m × n matrix, and let B and C be matrices of appropriate sizes.Then1. A(BC) = (AB)C2. A(B + C) = AB + AC3. (B + C)A = BA + CA4. r(AB) = (rA)B = A(rB) for any scalar r5. ImA = A = AInThree warnings.1. AB does not always equal BA.2. AB = AC does not necessarily imply that B = C.3. AB = 0 does not necessarily imply that A = 0 or B =