Linear Algebra

212 Chapter Three. Maps Between Spaceš 1.10 Prove each, assuming that the operations are defined, where G, H, and J arematrices, where Z is the zero matrix, and where r and s are scalars.(a) Matrix addition is commutative G + H = H + G.(b) Matrix addition is associative G + (H + J) = (G + H) + J.(c) The zero matrix is an additive identity G + Z = G.(d) 0 · G = Z(e) (r + s)G = rG + sG(f) Matrices have an additive inverse G + (−1) · G = Z.(g) r(G + H) = rG + rH(h) (rs)G = r(sG)1.11 Fix domain and codomain spaces. In general, one matrix can represent manydifferent maps with respect to different bases. However, prove that a zero matrixrepresents only a zero map. Are there other such matrices?̌ 1.12 Let V and W be vector spaces of dimensions n and m. Show that the spaceL(V, W) of linear maps from V to W is isomorphic to M m×n .̌ 1.13 Show that it follows from the prior questions that for any six transformationst 1 , . . . , t 6 : R 2 → R 2 there are scalars c 1 , . . . , c 6 ∈ R such that c 1 t 1 + · · · + c 6 t 6 isthe zero map. (Hint: this is a bit of a misleading question.)1.14 The trace of a square matrix is the sum of the entries on the main diagonal(the 1, 1 entry plus the 2, 2 entry, etc.; we will see the significance of the trace inChapter Five). Show that trace(H + G) = trace(H) + trace(G). Is there a similarresult for scalar multiplication?1.15 Recall that the transpose of a matrix M is another matrix, whose i, j entry isthe j, i entry of M. Verify these identities.(a) (G + H) trans = G trans + H trans(b) (r · H) trans = r · H tranš 1.16 A square matrix is symmetric if each i, j entry equals the j, i entry, that is, ifthe matrix equals its transpose.(a) Prove that for any H, the matrix H+H trans is symmetric. Does every symmetricmatrix have this form?(b) Prove that the set of n×n symmetric matrices is a subspace of M n×n .̌ 1.17 (a) How does matrix rank interact with scalar multiplication — can a scalarproduct of a rank n matrix have rank less than n? Greater?(b) How does matrix rank interact with matrix addition — can a sum of rank nmatrices have rank less than n? Greater?IV.2Matrix MultiplicationAfter representing addition and scalar multiplication of linear maps in the priorsubsection, the natural next map operation to consider is composition.2.1 Lemma The composition of linear maps is linear.Proof (This argument has appeared earlier, as part of the proof of Theo-