Linear Algebra/Sums and Scalar Products

Linear Algebra Sums and Scalar Products Matrix Operations Matrix Multiplication Recall that for two maps and with the same domain and codomain, the map sum has this definition. The easiest way to see how the representations of the maps combine to represent the map sum is with an example. Example 1.1 Suppose that are represented with respect to the bases and by these matrices. Then, for any represented with respect to , computation of the representation of gives this representation of . Thus, the action of is described by this matrix-vector product. This matrix is the entry-by-entry sum of original matrices, e.g., the entry of is the sum of the entry of and the entry of . Representing a scalar multiple of a map works the same way. Example 1.2 If is a transformation represented by then the scalar multiple map acts in this way. Therefore, this is the matrix representing . Definition 1.3def:SumScalarProdMats The sum of two same-sized matrices is their entry-by-entry sum. The scalar multiple of a matrix is the result of entry-by-entry scalar multiplication. Remark 1.4 These extend the vector addition and scalar multiplication operations that we defined in the first chapter. Theorem 1.5th:MatOpsRepMapOps Let be linear maps represented with respect to bases by the matrices and , and let be a scalar. Then the map is represented with respect to by , and the map is represented with respect to by . Proof Problem 2; generalize the examples above. A notable special case of scalar multiplication is multiplication by zero. For any map is the zero homomorphism and for any matrix is the zero matrix. Example 1.6 The zero map from any three-dimensional space to any two-dimensional space is represented by the zero matrix no matter which domain and codomain bases are used. Exercises Problem 1 Perform the indicated operations, if defined. <li> <li> <li> <li> <li> Problem 2exer:CorrspMapMatOps Prove Theorem 1.5. <li> Prove that matrix addition represents addition of linear maps. <li> Prove that matrix scalar multiplication represents scalar multiplication of linear maps. Problem 3 Prove each, where the operations are defined, where , , and are matrices, where is the zero matrix, and where and are scalars. <li> Matrix addition is commutative . <li> Matrix addition is associative . <li> The zero matrix is an additive identity . <li> <li> <li> Matrices have an additive inverse . <li> <li> Problem 4 Fix domain and codomain spaces. In general, one matrix can represent many different maps with respect to different bases. However, prove that a zero matrix represents only a zero map. Are there other such matrices? Problem 5exer:LinMapsIsoMatSp Let and be vector spaces of dimensions and . Show that the space of linear maps from to is isomorphic to . Problem 6 Show that it follows from the prior questions that for any six transformations there are scalars such that is the zero map. (Hint: this is a bit of a misleading question.) Problem 7trace The trace of a square matrix is the sum of the entries on the main diagonal (the entry plus the entry, etc.; we will see the significance of the trace in Chapter Five). Show that . Is there a similar result for scalar multiplication? Problem 8transpose Recall that the transpose of a matrix is another matrix, whose entry is the entry of . Verifiy these identities. <li> <li> Problem 9symmetric A square matrix is symmetric if each entry equals the entry, that is, if the matrix equals its transpose. <li> Prove that for any , the matrix is symmetric. Does every symmetric matrix have this form? <li> Prove that the set of symmetric matrices is a subspace of . Problem 10 <li> How does matrix rank interact with scalar multiplication— can a scalar product of a rank matrix have rank less than ? Greater? <li> How does matrix rank interact with matrix addition— can a sum of rank matrices have rank less than ? Greater? /Solutions/ Linear Algebra Sums and Scalar Products Matrix Operations Matrix Multiplication