Page 1 of 5
Math 430
Siddharth Dahiya
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August 26, 2011
Problem 1.2.7
1.2.7. Consider the matrices
1 1 3
A = 1 4 2 B =
3
0
6
6 0 3
4 2 1
2
3
C = 3 4
1
2
Compute the indicated combinations where possible.
(a) 3A - B
This is not possible since
Page 1 of 5
Math 430
Siddharth Dahiya
[email protected]
December 3, 2011
Homework # 13
7.1.28. True or false : The set of linear transformations L : R2 R2 such that
L
1
0
0
0
=
forms a subspace of L(R2 , R2 ). If true, what is its dimension?
True.
Ax = b
A=
Page 1 of 6
Math 430
Siddharth Dahiya
[email protected]
November 23, 2011
Homework #
7.1.2. Which of the following functions F : R2 R2 are linear?
(a) F
x
y
=
xy
x+y
Linear.
1. Distributive
F
x+
y+
=
(x y ) + ( )
(x + y ) + ( + )
=
xy
+
x+y
+
=F
2. Scalar M
Page 1 of 4
Math 430
Siddharth Dahiya
[email protected]
November 12, 2011
Homework # 11
5.1.5. Find all values of a so that the vectors
a
a
,
1
1
form an orthogonal basis of R2 under:
(a) The dot product
Under dot product,
a
a
1
1
=0
(a)(a) + (1)(1) = 0
a
Page 1 of 6
Math 430
Siddharth Dahiya
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November 7, 2011
Homework # 10
3.6.16(a). Use the formula e2i = (ei )2 to deduce the well-known trigonometric identities for cos 2 and sin 2 .
Using Eulers Formulae, ei = cos + i sin ,
(ei )2 = (cos +
Page 1 of 6
Math 430
Siddharth Dahiya
[email protected]
October 24, 2011
Homework # 9
3.2.36. Use the L2 inner product
1
f, g =
f (x)g(x)dx
1
to answer the following:
(a) Find the angle between the functions 1 and x. Are they orthogonal?
1
1, x =
(1)(x)dx =
Page 1 of 8
Math 430
Siddharth Dahiya
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October 30, 2011
Homework # 8
3.1.3. Show that v, w = v1 w1 + v1 w2 + v2 w1 + v2 w2 does not dene an inner product on R2 .
By Inner Product Axioms, the inner product has to be positive denite i.e.
v ,
Math 430
Siddharth Dahiya
[email protected]
October 14, 2011
Homework # 7
3.1.3. Show that v, w = v1 w1 + v1 w2 + v2 w1 + v2 w2 does not dene an inner product on R2 .
Page 1 of 6
Math 430
Siddharth Dahiya
[email protected]
October 14, 2011
Homework # 7
Page 2
Page 1 of 6
Math 430
Siddharth Dahiya
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September 25, 2011
Homework # 6
2.3.3. Follow the instructions.
1
1
0
(a) Determine whether 2 is in the span of 1 and 1
3
0
1
Here,
1
1
0
c = 2 ; v1 = 1 ; v2 = 1
3
0
1
Then, for c to be in the span of
Page 1 of 8
Math 430
Siddharth Dahiya
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September 16, 2011
Homework # 4
2.1.2. Show that the positive quadrant
Q = cfw_(x, y )|x, y > 0 R2
forms a vector space if we dene addition by (x1 , y1 ) + (x2 , y2 ) = (x1 x2 , y1 y2 ) and scalar mult
Page 1 of 4
Math 430
Siddharth Dahiya
[email protected]
September 12, 2011
Homework # 3
13. Let A and B be m n matrices.
(a) Suppose that vT Aw = vT B w for all vectors v, w. Prove that A = B
Let v and w be any vectors.
Given,
vT Aw = vT B w
For this mutlip
Page 1 of 5
Math 430
Siddharth Dahiya
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September 2, 2011
Homework # 2
1. A square matrix A is called nilpotent if Ak = O for some k 1.
0
(a) Show that A = 0
0
0
0
A=
0
0
2 = 0
AA=A
0
0
A2 A = A3 = 0
0
12
0 1 is nilpotent.
00
12
0 1
00
01
0
Page 1 of 9
Math 430
Siddharth Dahiya
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December 9, 2011
Homework # 14
7.2.29. Suppose a linear transformation L : Rn Rn is represented by a symmetric matrix with respect to
the standard basis e1 , . . . , en
(a) Prove that its matrix repres