Linear Algebra, 2020a

244 Chapter Three. Maps Between Spaces
̌ 2.34 Represent the identity transformation id: V → V with respect to B, B for any
basis B. Thisistheidentity 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).
2.35 (a) Prove that for any 2×2 matrix T there are scalars c 0 ,...,c 4 that are not
all 0 such that the combination c 4 T 4 + c 3 T 3 + c 2 T 2 + c 1 T + c 0 I is the zero matrix
(where I is the 2×2 identity matrix, with 1’s in its 1, 1 and 2, 2 entries and zeroes
elsewhere; see Exercise 34).
(b) Let p(x) be a polynomial p(x) =c n x n + ··· + c 1 x + c 0 . If T is a square
matrix we define p(T) to be the matrix c n T n + ···+ c 1 T + c 0 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
the minimal polynomial of this matrix.
(√ )
3/2
√ −1/2
1/2 3/2
(This is the representation with respect to E 2 , E 2 , the standard basis, of a rotation
through π/6 radians counterclockwise.)
2.36 The infinite-dimensional space P of all finite-degree polynomials gives a memorable
example of the non-commutativity of linear maps. Let d/dx: P → P be the
usual derivative and let s: P → P be the shift map.
a 0 + a 1 x + ···+ a n x n
s
↦−→ 0 + a 0 x + a 1 x 2 + ···+ a n x n+1
Show that the two maps don’t commute d/dx ◦ s ≠ s ◦ d/dx; in fact, not only is
(d/dx ◦ s)−(s ◦ d/dx) not the zero map, it is the identity map.
2.37 Recall the notation for the sum of the sequence of numbers a 1 ,a 2 ,...,a n .
n∑
a i = a 1 + a 2 + ···+ a n
i=1
In this notation, the i, j entry of the product of G and H is this.
r∑
p i,j = g i,k h k,j
k=1
Using this notation,
(a) reprove that matrix multiplication is associative;
(b) reprove Theorem 2.7.
IV.3
Mechanics of Matrix Multiplication
We can consider matrix multiplication as a mechanical process, putting aside for
the moment any implications about the underlying maps.