Linear Algebra, 2020a

Section IV. Matrix Operations 235
(e) 3
( ) ( )
2 1 1 1 4
+ 2
3 0 3 0 5
1.9 Give the matrix representing the zero map from R 4 to R 2 , with respect to the
standard bases.
1.10 Prove Theorem 1.4.
(a) Prove that matrix addition represents addition of linear maps.
(b) Prove that matrix scalar multiplication represents scalar multiplication of
linear maps.
̌ 1.11 Prove each, assuming that the operations are defined, where G, H, and J are
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)
1.12 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?
̌ 1.13 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 M m×n .
̌ 1.14 Show that it follows from the prior question that for any six transformations
t 1 ,...,t 6 : R 2 → R 2 there are scalars c 1 ,...,c 6 ∈ R such that not every c i equals 0
but c 1 t 1 + ···+ c 6 t 6 is the zero map. (Hint: the six is slightly misleading.)
1.15 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?
1.16 Recall that the transpose of a matrix M is another matrix, whose i, j entry is
the j, i entry of M. Verify these identities.
(a) (G + H) T = G T + H T
(b) (r · H) T = r · H T
̌ 1.17 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 square H, the matrix H + H T is symmetric. Does every
symmetric matrix have this form?
(b) Prove that the set of n×n symmetric matrices is a subspace of M n×n .
̌ 1.18 (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?