MATH 223, Linear Algebra Fall 2012

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MATH 223, Linear Algebra Fall 2012

Assignment 6, Due in class Friday November 16

y1

1. Find the general solution to the following system of differential equations: y2

y3

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



=

=

=



−12y1 + 13y2 + 10y3

4y1 − 3y2 − 4y3

−21y1 + 21y2 + 19y3



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.)





20

2

0 , show that T is a projection.

(a) If T = TA is represented by the matrix A =  0 1

−1 0 −1

(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 minimal polynomials of each of the following matrices over the real

and







5

51

2 10

300

2 1 .

numbers.  0 3 0 ,  −1 4 0 ,  0

−1 −2 2

−1 1 3

003

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 = C n . Show that v , w = v · w is not an inner product on V .

(c) Now let V = C 3 , 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

2



f, g =



tf (t)g (t)dt.

1



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

2



2



t2 f (t)dt



≤



1



1



7

3



2



tf (t)2 dt .

1