Linear Algebra

Section IV. Matrix Operations 211These operations extend the first chapter’s addition and scalar multiplicationoperations on vectors.1.4 Theorem Let h, g: V → W be linear maps represented with respect to basesB, D by the matrices H and G, and let r be a scalar. Then the map h + g: V → Wis represented with respect to B, D by H + G, and the map r · h: V → W isrepresented with respect to B, D by rH.Proof Exercise 9; generalize the examples above.QED1.5 Remark Recall Remark III.1.6 following Theorem III.1.4. That theorem saysthat matrix-vector multiplication represents the application of a linear map andthe remark notes that the theorem simply justifies the definition of matrix-vectormultiplication. In some sense the theorem has to hold, because if it didn’t thenwe would adjust the definition to make the theorem hold. The above theorem isanother example of such a result; it shows that our definition of the operationsis sensible.A special case of scalar multiplication is multiplication by zero. For any map0 · h is the zero homomorphism and for any matrix 0 · H is the matrix with allentries zero.1.6 Definition A zero matrix has all entries 0. We write Z n×m or simply Z(another common notation is 0 n×m or just 0).1.7 Example The zero map from any three-dimensional space to any twodimensionalspace is represented by the 2×3 zero matrix( )0 0 0Z =0 0 0no matter what domain and codomain bases we use.Exerciseš 1.8 Perform(the indicated) (operations,)if defined.5 −1 2 2 1 4(a)+6 1 1 3 0 5( )2 −1 −1(b) 6 ·1 2 3( ) ( )2 1 2 1(c) +0 3 0 3( ) ( )1 2 −1 4(d) 4 + 53 −1 −2 1( ) ( )2 1 1 1 4(e) 3 + 23 0 3 0 51.9 Prove Theorem 1.4.(a) Prove that matrix addition represents addition of linear maps.(b) Prove that matrix scalar multiplication represents scalar multiplication oflinear maps.