MATH 223, Linear Algebra Fall 2012

Student name: Student number:

Use this page as a cover page for your assignment

MATH 223, Linear Algebra Fall 2012 Assignment 6, Due in class Friday November 16

1. Find the general solution to the following system of differential equations: y′1 = −12y1 + 13y2 + 10y3 y′2 = 4y1 − 3y2 − 4y3 y′3 = −21y1 + 21y2 + 19y3

(Hint: one eigenvalue of the corresponding matrix is λ1 = 1.)

2. A linear operator T on a vector space V is called a projection if T 2 = T . (We will be looking at orthogonal projections later.)

(a) If T = TA is represented by the matrix A =

  2 0 20 1 0 −1 0 −1

 , show that T is a projection.

(b) Show that, if T is a projection on V , then so is I −T; here I is the identity operator on V .

(c) Show that, if T is a projection on, then V = ker(T) ⊕ Im(T). (d) Show that, if T is a projection on V , its only possible eigenvalues are 0 and 1. (e) Show that, if T is a projection on V and V is finite-dimensional, then T is diagonalizable.

3. Find the characteristic and minimal polynomials of each of the following matrices over the real

numbers.

  3 0 00 3 0

0 0 3

 ,   2 1 0−1 4 0 −1 1 3

 ,   5 5 10 2 1 −1 −2 2

 .

4. (a) Let V = R4 with its usual inner product 〈v,w〉 = v·w. Let u = (1, 1, 1, 1), v = (2, 3,−1, 2), and w = (3, 4, 0, 3). Determine each of

||u||, ||v||, ||w||,〈u,v〉, 〈u,w〉, 〈v,w〉.

(b) Let V = Cn. Show that 〈v,w〉 = v ·w is not an inner product on V . (c) Now let V = C3, equipped with its usual inner product 〈v,w〉 = v · w. Set u = (1 +

i, 2,−3 − i) and v = (i, 3i, 5 − 2i). Find

||u||, ||v||, 〈u,v〉.

(d) Let V be any real inner product space, with inner product 〈·, ·〉. Prove that for all u,v ∈ V

〈u + v,u−v〉 = ||u||2 −||v||2.

5. Let V = Mn(R) be the real vector space of n × n real matrices, let U be the subspace of V consisting of upper triangular matrices, and let W be the subspace of U consisting of diagonal matrices. For any two matrices A,B ∈ V , let

〈A,B〉 = tr(AB).

Show that 〈·, ·〉 is not an inner product on V , but that its restriction to W is an inner product on W . Is the restriction of 〈·, ·〉 to U an inner product on U? Justify your answer.

6. Let V be the real vector space of continuous real-valued functions on the interval [1, 2], and for any f,g ∈ V let

〈f,g〉 = ∫ 2

1

tf(t)g(t)dt.

Show that this defines an inner product on V , and that for any f ∈ V we have(∫ 2 1

t2f(t)dt )2 ≤

7 3

(∫ 2 1

tf(t)2dt ) .

1