Linear Algebra

Section IV. Matrix Operations 213
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2 1 2 1
(c) +
0 3 0 3
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1 2 −1 4
(d) 4 +5
3 −1 −2 1
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2 1 1 1 4
(e) 3 +2
3 0 3 0 5
1.8 Prove Theorem 1.5.
(a) Prove that matrix addition represents addition of linear maps.
(b) Prove that matrix scalar multiplication represents scalar multiplication of
linear maps.
� 1.9 Prove each, where the operations are defined, where G, H, andJare matrices,
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)
1.10 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?
� 1.11 Let V and W be vector spaces of dimensions n and m. Show that the space
L(V,W) of linear maps from V to W is isomorphic to Mm×n.
� 1.12 Show that it follows from the prior questions that for any six transformations
t1,...,t6 : R 2 → R 2 there are scalars c1,...,c6 ∈ R such that c1t1 + ···+ c6t6 is
the zero map. (Hint: this is a bit of a misleading question.)
1.13 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 in
Chapter Five). Show that trace(H + G) =trace(H)+trace(G). Is there a similar
result for scalar multiplication?
1.14 Recall that the transpose of a matrix M is another matrix, whose i, j entry is
the j, i entry of M. Verifiy these identities.
(a) (G + H) trans = G trans + H trans
(b) (r · H) trans = r · H trans
� 1.15 A square matrix is symmetric if each i, j entry equals the j, i entry, that is, if
the matrix equals its transpose.
(a) Prove that for any H, the matrix H + H trans is symmetric. Does every
symmetric matrix have this form?
(b) Prove that the set of n×n symmetric matrices is a subspace of Mn×n.
� 1.16 (a) How does matrix rank interact with scalar multiplication—can a scalar
product 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 n
matrices have rank less than n? Greater?