Vector Spaces 2003 (CD Burn)

7/31/2019 Vector Spaces 2003 (CD Burn)

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TMA 2023 / SMA 3013

Semester 2010/ 2011

Assignment

Vector Spaces

No. Name Student ID

1 NAJWA HANIM BINTI YUSOFF D20091034560

2 NUR AMANINA NAJIHAH BINTI AB

LATIB

D20091034561

3 NOR ATIQAH BINTI FAHRUL RADZI D20091034569

4 NUR FATIHAH BINTI MOHD KARIM D20091034574

5 JENIFER JOSLI D20091034575


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VECTOR SPACES

Definition:

A set V is called a vector space over the real numbers provided that there are two operations-

addition, denoted by + , and scalar multiplication, denoted by - that satisfy all the following

axioms. The axioms must hold for all vectors u, v, w in V and all scalars c and d R.

Vector addition: This assigns to any u, v V a sum u + v in V Scalar multiplication: This assign to any u V, k K, a product ku V.

These are the 10 axioms in Vector Space:

1. u + vV( Vis closed under addition)2. u + v = v + u (commutative law)3. (u + v ) + w = u (v + w) (associative law)4. There exists a vector Ov, V where u + Ov = u = Ov + u, all elements in u ( the

existing of a zero vector)

5. For each vector u V, there exists a vector u V where u + (-u) = Ov. (existinginverse of addition or a negative vector).

6. c uV(Vis closed under multiplication)7. c (u +v) = cu + cv (left distributive law)8. (c+ d) u= cu + du9. c(du) = (cd)u10.I u = u

In this section, we use the special symbols + and of the previous definition to distinguish

vector addition and scalar multiplication from ordinary addition and multiplication of real

numbers (R).

Euclidean Vector Spaces

The set V = R with the standard operations of addition and scalar multiplication is a vector

space.


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Vector Spaces of Matrices

Show that the set V = M mxn of all m x n matrices is a vector space over the scalar field R,

with + and defined componentwise.

Since addition of matrices is componentwise, the sum of two m x n matrices is another m

x n matrix as is a scalar times an m x n matrix. Thus, the closure axioms (axioms 1 and 6) are

satisfied.

Exercises

1. Determine whether these matrices are vector spaces or not.A = , B =

i. A + BV( Vis closed under addition)A + B =

=

ii. A + B = B + A (commutative law)LHS A + B =

=

RHS B + A =

=

=

iii. (A + B ) + C = A + (B + C) (associative law) which is C =LHS (A + B) + C =

=

=


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RHS A + (B + C) =

=

=

iv. A + 0= A = 0 + A

v. A + (-A) = 0

=

vi. cAV(Vis closed under multiplication) which is c = x, x is a scalar.

vii. c(A + B) = cA + cB (left distributive law)LHS c(A + B) =

=

=

=

RHS cA + cB =

=

=


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viii. (c + d) A= cA + dA, which is d = y, y is a scalar.LHS (c + d) A =

=

=

RHS cA + dA =

=

=

ix. c(dA) = (cd)ALHS c(dA) =

=

=

RHS (cd)A =

=

x. IA = A, which is I =

=


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=

Therefore, the 10 axioms are satisfied. Thus, these matrices are vector spaces.

2. Let V = R. Show that this addition A + B = 3A + 3B is commutative but notassociative.

i. To show whether + is commutative.A + B = B + A

A + B = 3A + 3B

= 3 (A + B)

= 3 (B + A) commutative

= B + A

This operation is commutative.

ii. To show whether + is associative.(A + B) + C = A + (B + C)

LHS (A + B) + C = (3A + 3B) + C

= 3(3A + 3B) + 3C

= 9A + 9B + 3C

RHS A + (B + C) = A + (3B + 3C)

= 3A + 3(3B + 3C)

= 3A + 9B + 9C

This operation is not associative.

Thus, V is not a vector spaces.

Counter example:

Let A = 1, B = 2 and C = 3.

LHS (1 + 2) + 3 = [3(1) + 3(2)] + 3

= (3 + 6) + 3


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= 9 + 3

= 3(9) + 3(3)

= 27 + 9

= 36

RHS 1 + (2 + 3) = 1 + [3(2) + 3(3)]

= 1 + (6 + 9)

= 1 + 15

= 3(1) + 3(15)

= 3 + 45

= 48

LHS RHS.

Thus, it is not a vector spaces.

3. Show that the operation is a vector space or not.i) To show whether + is a commutative

A + B = B + A

A + B = 5A + 5B

= 5(A + B)

= 5(B + A) commutative

= B + A

Therefore, the operation is commutative

ii) To show whether + is associative(A + B) + C = A + (B + C)

LHS : (A + B) + C = (5A + 5B) + C

= 5 (5A + 5B) + 5C

= 25A + 25B + 5C

RHS : A + (B + C) = A + (5B + 5C)

= 5A + 5(5B + 5C)


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= 5A + 25B + 25C

LHS RHS Thus, V is not a vector space.

Counter example:

Let A = 3, B = 6, C = 9

LHS : (3 + 6) + 9 = [5(3) + 5(6)] + 9

= [15 + 30] + 9

= 45 + 9

= 5(45) + 5(9)

= 270

RHS : 3 + (6 + 9) = 3 + [5(6) + 5(9)]

= 3 + [30 + 45]

= 3 + 75

= 5(3) + 5(75)

= 390

LHS RHS Thus, it is not a vector space.

4. Given A = , B = , C =Show that this operation from two matrices is a vector space or not.

i) To show whether + is commutative: A + B = B + ALHS: A + B = 3 + 3

= +

=

RHS: B + A = 3 + 3

= +


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=

Thus, the operation is commutative.

ii) To show whether + is associative: (A + B) + C = A + (B + C)LHS: (A + B) + C = ( + ) +

= (3 + 3 ) +

= ( + ) +

= 3 + 3

= +

=

RHS: A + (B + C) = + ( + )

= + (3 + 3 )

= + ( + )

= 3 + 3

= +

=

The operation is not associative. Thus, this is not a vector space.Counter example: Given A = , B = , C =

LHS: (A + B) + C = ( + ) +


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= (3 + 3 +

= ( + ) +

= 3 + 3

= +

=

RHS: A + (B + C) = + ( +

= + (3 + 3 )

= + ( + )

= +

= 3 + 3

= +

=

LHS RHS Thus, this is not a vector space.


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REFERENCES

Bronson, R., Costa, G, B. (2007). Linear Algebra: an Introduction. Elsevier Inc. pp: 8597.

Cullen, C. (1997).Linear algebra with application. Addison-Wesley. pp: 136 - 139.

Lipschutz, S., Lipson, M, L. (2001). Theory and Problems of Linear Algebra(3). McGRAW-

HILL International Edition. pp:

Lipschutz, S., Lipson, M, L. (2009). Linear Algebra(4). McGRAW-HILL International

Edition.


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Group discussion / Meeting report

No Date Activities Task Assign

1 5 March 2011 Group discussion at library.

Finding reference books

From this discussion, we

have found out the

definition and also the 10

axioms in Vector Spaces.

2 9 March 2011 Preparing for the Vector

Spaces module at library.

From this discussion, we

have prepared the module of

Vector Spaces.

3 12 March 2011 Prepare the powerpoint and

ready for the presentation.

From this group discussion,

we have prepared for the

presentation slides in VectorSpaces.

Vector Spaces 2003 (CD Burn)

Documents

Transcript of Vector Spaces 2003 (CD Burn)