Linear Algebra

220 Chapter 3. Maps Between Spaces
� 2.24 (a) How does matrix multiplication interact with scalar multiplication: is
r(GH) =(rG)H? IsG(rH) =r(GH)?
(b) How does matrix multiplication interact with linear combinations: is F (rG+
sH) =r(FG)+s(FH)? Is (rF + sG)H = rFH + sGH?
2.25 We can ask how the matrix product operation interacts with the transpose
operation.
(a) Show that (GH) trans = H trans G trans .
(b) A square matrix is symmetric if each i, j entry equals the j, i entry, that is,
if the matrix equals its own transpose. Show that the matrices HH trans and
H trans H are symmetric.
� 2.26 Rotation of vectors in R 3 about an axis is a linear map. Show that linear
maps do not commute by showing geometrically that rotations do not commute.
2.27 In the proof of Theorem 2.12 some maps are used. What are the domains and
codomains?
2.28 How does matrix rank interact with matrix multiplication?
(a) Can the product of rank n matrices have rank less than n? Greater?
(b) Show that the rank of the product of two matrices is less than or equal to
the minimum of the rank of each factor.
2.29 Is ‘commutes with’ an equivalence relation among n×n matrices?
� 2.30 (This will be used in the Matrix Inverses exercises.) Here is another property
of matrix multiplication that might be puzzling at first sight.
(a) Prove that the composition of the projections πx,πy : R 3 → R 3 onto the x
and y axes is the zero map despite that neither one is itself the zero map.
(b) Prove that the composition of the derivatives d 2 /dx 2 ,d 3 /dx 3 : P4 →P4 is
the zero map despite that neither is the zero map.
(c) Give a matrix equation representing the first fact.
(d) Give a matrix equation representing the second.
When two things multiply to give zero despite that neither is zero, each is said to
be a zero divisor.
2.31 Show that, for square matrices, (S + T )(S − T ) need not equal S 2 − T 2 .
� 2.32 Represent the identity transformation id: V → V with respect to B,B for any
basis B. This is the identity matrix I. Show that this matrix plays the role in
matrix multiplication that the number 1 plays in real number multiplication: HI =
IH = H (for all matrices H for which the product is defined).
2.33 In real number algebra, quadratic equations have at most two solutions. That
is not so with matrix algebra. Show that the 2×2 matrix equation T 2 = I has
more than two solutions, where I is the identity matrix (this matrix has ones in
its 1, 1and2, 2 entries and zeroes elsewhere; see Exercise 32).
2.34 (a) Prove that for any 2×2 matrixT there are scalars c0,...,c4 such that the
combination c4T 4 + c3T 3 + c2T 2 + c1T + I isthezeromatrix(whereI is the 2×2
identity matrix, with ones in its 1, 1and2, 2 entries and zeroes elsewhere; see
Exercise 32).
(b) Let p(x) be a polynomial p(x) =cnx n + ··· + c1x + c0. If T is a square
matrix we define p(T ) to be the matrix cnT n + ···+ c1T + I (where I is the
appropriately-sized identity matrix). Prove that for any square matrix there is
a polynomial such that p(T ) is the zero matrix.
(c) The minimal polynomial m(x) of a square matrix is the polynomial of least
degree, and with leading coefficient 1, such that m(T ) is the zero matrix. Find