Linear Algebra Paper
of 4
-
Upload
ayub-faridi -
Category
Documents
-
view
218 -
download
0
Transcript of Linear Algebra Paper
-
8/3/2019 Linear Algebra Paper
1/4
Linear Algebra
Exercises 1
1.1 Let A = [aij]45, B = [bij]45, C = [cij]52, D = [dij]42, and E = [eij]54. Determine
which of the following matrix expressions are defined. For those which are defined,
give the size of the resulting matrix.
(a) BA (b) AC+ D (c) AE+ B (d) AB + B (e) E(A + B)
(f) E(AC) (g) ETA (h) (AT + E)D.
1.2 Consider the matrices
A =
3 01 2
1 1
B =
4 10 2
C =
1 4 2
3 1 5
D =
1 5 21 0 1
3 2 4
E =
6 1 31 1 2
4 1 3
.
Compute the following (where possible).
(a) 4E 2D (b) 2B C (c) AB (d) BA (e) (3E)D(f) (AB)C (g) A(BC) (h) (DA)T (i) ATDT (j) tr(4ET D)(k) tr(CCT).
1.3 Solve the following matrix equation for a,b,c and d:a b b + c
3d + c 2a 4d
=
8 1
7 6
.
1.4 Let
A =
3 1
5 2
, B =
2 34 4
, C =
2 0
0 3
.
Verify that (AB)1 = B1 A1 and (ABC)1 = C1 B1 A1.
1.5 Let D be a non-singular matrix such that
(7D)1 =
3 7
1 2
.
Find D.
-
8/3/2019 Linear Algebra Paper
2/4
1.6 Evaluate tr(0nn) and tr(In), and prove that
(a) tr(A + B) = tr(A) + tr(B),
(b) tr(AB) = tr(BA),
(c) tr(A) = tr(A)
for any n n matrices A, B and R.
1.7 Which of the following are elementary matrices?
(a)
1 0
5 1
(b)
5 1
1 0
(c)
1 0
0
3
(d)
0 0 1
0 1 0
1 0 0
(e)
1 1 0
0 0 1
0 0 0
(f)
1 0 0
0 1 9
0 0 1
(g)
2 0 0 2
0 1 0 0
0 0 1 0
0 0 0 1
.
1.8 Consider the matrices
A = 3 4 12 7 1
8 1 5
B = 8 1 5
2 7 13 4 1
C = 3 4 1
2 7 12 7 3
.
Find elementary matrices E1, E2, E3 and E4 such that
(a) E1 A = B (b) E2 B = A (c) E3 A = C (d) E4 C = A.
1.9 Premultiply the general 4 4 matrix A = [aij]44 by each of
P1 =
1 0 0 0
c1
c2
c3
c4
0 0 1 0
0 0 0 1
, P2 = 1 0 c1 0
0 1 c2
00 0 c3 0
0 0 c4 1
and describe the effect on the rows. Hence express each Pi as a product of four
elementary matrices.
-
8/3/2019 Linear Algebra Paper
3/4
1.10 In each of the following, determine whether the matrix is in echelon form, reduced
echelon form, both or neither.
(a)
1 2 0 3 0
0 0 1 1 0
0 0 0 0 10 0 0 0 0
(b)
1 0 0 5
0 0 1 30 1 0 4
(c)
1 0 3 10 1 2 4
(d)
1 7 5 50 1 3 2
(e)
1 3 0 2 0
1 0 2 2 0
0 0 0 0 1
0 0 0 0 0
(f)
0 00 0
0 0
.
1.11 Reduce the following matrices to reduced echelon form, determining in each case the
appropriate premultiplying matrix P.
(a)
1 1 13 1 2
5 1 3
(b)
1 3 4 31 2 2 2
1 6 4 1
(c)
1 1 1
1 2 4
1 3 9
1 4 16
(d)
1 2 1 2
2 4 3 4
1 3 2 3
0 3 1 3
(e)
2 2 31 1 0
1 2 1
.
1.12 In each of the following, suppose that the augmented matrix for a system of linear
equations has been reduced by row operations to the given echelon form and hence
solve the system.
(a)
1 3 4 | 70 1 2 | 2
0 0 1 | 5
(b)
1 0 8 5 | 60 1 4 9 | 3
0 0 1 1 | 2
(c)
1 7 2 0 8 | 3
0 0 1 1 6 | 50 0 0 1 3 | 90 0 0 0 0 | 0
(d) 1
3 7
|1
0 1 4 | 00 0 0 | 1
.
-
8/3/2019 Linear Algebra Paper
4/4
1.13 Find the general solutions (if they exist) of the following systems of equations.
(a) 2x1 + x2 + 3x3 = 0
3x1 2x2 + x3 = 0x1 3x2 2x3 = 0
(b) 2x1 + 3x2 + 4x3 = 1
5x1 + 6x2 + 7x3 = 2
8x1 + 9x2 + 10x3 = 4
(c) x1 3x2 + 5x3 7x4 = 22x1 + 4x2 6x3 + 8x4 = 2
x1 + x2 + x3 + x4 = 2
x1 + 5x2 + 2x3 + 5x4 = 7
(d) 3x1 + x2 + 2x3 + 4x4 = 3
5x1 + 2x2 + 3x3 + 6x4 = 5
4x1 + x2 + 3x3 + 6x4 = 4
5x1 + x2 + 4x3 + 8x4 = 5
(e) 2x1 + x2 + 3x3 + 5x4 = 6
3x1 + 2x2 + 4x3 + 6x4 = 8
x1 + 3x2 + 2x3 + 7x4 = 3
1.14 Letx
= [1, 0, 2, 2]
T
,y
= [3, 6, 2, 0]
T
. Calculate
x, y, 4x, 9y, x + y.
Verify that x+y x+y and show that the angle between x and y is cos1(1/3).
1.15 Find the Euclidean inner product x y when
(a) x = [3, 1, 4, 5]T, y = [2, 2, 4, 3]T
(b) x = [1, 1, 0, 4, 3]T, y = [2, 2, 0, 2, 1]T.
1.16 Let the distance between x and y in Rn be defined by d(x,y) = x y. Show that(a) d(x,y) 0, (b) d(x,y) = 0 iff x = y (c) d(x,y) = d(y,x)(d) d(x,z) d(x,y) + d(y,z).
Find the distance between [1, 2, 1, 4, 7, 3]T and [2, 1, 3, 5, 4, 5]T in R6.
1.17 For x,y Rn, prove the following:-
(a) x y iff x2 + y2 = x + y2
(b) x y2 = x2 + y2 2x y cos ( the angle between x and y)(c) x + y2 + x y2 = 2x2 + 2y2.
To what theorems of Euclidean geometry do these results correspond in the case of
R2?
