Linear Algebra
Course Teacher: Prof. Dr. Khandaker Farid Uddin Ahmed
Textbooks:
1. Elementary Linear Algebra: Applications Version (9th edition) by
Howard Anton and Chris Rorres
2. Introduction to Linear Algebra by Gilbert Strang (MIT)
3. Precalculus- Sullivan
Linear system of equations:
Consider a system of m linear equations in the n unknowns
a11x1 + a12x2 + …… + a1nxn = b1
a21x1 + a22x2+ …. + a2nxn = b2
…..
…..
am1x1 + am2 x2 + …. + amnxn = bm
aij εℜ, f or i = 1, m and j = 1, n
i= row, j = column
X = (x1, x2 , ….. , xn) = vector of n components
A = coefficient matrix
AX = B
Typed by Ridwan Abrar, BUET EEE ‘16
,Chapter 5
Real
Vector spaces and subspaces (Chapter 5)
Definition (Vector space)
Let K be a given field of scalars (k εℜ) = {x :− ∞ < x < ∞} = (− ∞, ∞)
(a,b), a<b
and let V be an arbitrary nonempty set of objects with sum and scalar
multiplication which assigns to each u, v εV , a sum u+v ε V ; and for
any k ε K and for any u ε V , the product ku ε V . Then V is a vector space
and the objects of V are vectors if V satisfies the following conditions:
For any u, v, w ε V , and for any k, k1, k2 ε k , we have:
i) u + (v+w) = (u+v) + w
ii) u+ v = v + u
iii) For any u ε V , there exists a 0 ε V such that 0 + u = u
iv) For any u ε V , there exists a − u ε V such that u + (-u) = 0
v) k(u+v) = ku+kv;
vi) (k1+k2)u = k1u + k2u;
vii) (k1k2)u = k1(k2u)
viii) 1u = u
Vector spaces and subspaces
Definition:
k ε ℜ => V is a real vector space
k ε ⊂ => V is a complex vector space
Typed by Ridwan Abrar, BUET EEE ‘16
,Examples:
1. All lines passing through the origin such as ax+by = 0 and ax+by+c
=0
3D = all planes passing through the origin
In case of 3D, a , b and c are direction ratios
2.
3. Set of all continuous functions
Linear Combination:
The vector u1, u2, …., un form a set S = { u1, u2, ….., un} for a vector space
V. Then any non-zero vector u ε V is a linear combination of the
vectors in S if u = c1u1 + c2u2 + c3u3 + …. + cnun
for any ci ε K , i = 1, n
Linear dependence:
The vectors u1, u2, …, un are linearly dependent if we can find scalars c1,
c2, ….. cn , not all of them zero such that c1u1+c2u2+...+cnun = 0
Example 1:
︿ ︿ ︿
Every vector in ℜ3 is a linear combination of i, j and k.
i = (1,0,0) = column of matrix = e1 =
j = (0,1,0) = e2
k = (0,0,1) = e3
Set u = (a,b,c) ε ℜ3
Then u = (a,b,c) = ae1 + be2 + ce3
= a(1,0,0) + b(0,1,0) + c(0,0,1)
Typed by Ridwan Abrar, BUET EEE ‘16
, Example 2
Examine whether the vectors w = (9,2,7) is a linear combination of u =
(1,2,-1) and v = (6,4,2).
Solution:
Take any scalars x,y such that
w = xu + yv
=> (9,2,7) = x(1,2,-1) + y(6,4,2) = (x,2x,-x) + (6y,4y,2y)
=> (9,2,7) = (x+6y, 2x+4y, -x+2y)
x + 6y = 9
2x+ 4y = 2
-x + 2y = 7
x = -3 and y = 2
(unique solution)
Non homogeneous system = None of the constants are zero
Consistent system = has a solution
w = -3u + 2v
Thus w is a linear combination of u and v.
Example 3
Examine whether w = (4,1,-8) is a linear combination of u, v.
w = xu + yv
=> (4,-1,8) = (x+6y, 2x+4y, -x+2y)
The system is:
Typed by Ridwan Abrar, BUET EEE ‘16
Course Teacher: Prof. Dr. Khandaker Farid Uddin Ahmed
Textbooks:
1. Elementary Linear Algebra: Applications Version (9th edition) by
Howard Anton and Chris Rorres
2. Introduction to Linear Algebra by Gilbert Strang (MIT)
3. Precalculus- Sullivan
Linear system of equations:
Consider a system of m linear equations in the n unknowns
a11x1 + a12x2 + …… + a1nxn = b1
a21x1 + a22x2+ …. + a2nxn = b2
…..
…..
am1x1 + am2 x2 + …. + amnxn = bm
aij εℜ, f or i = 1, m and j = 1, n
i= row, j = column
X = (x1, x2 , ….. , xn) = vector of n components
A = coefficient matrix
AX = B
Typed by Ridwan Abrar, BUET EEE ‘16
,Chapter 5
Real
Vector spaces and subspaces (Chapter 5)
Definition (Vector space)
Let K be a given field of scalars (k εℜ) = {x :− ∞ < x < ∞} = (− ∞, ∞)
(a,b), a<b
and let V be an arbitrary nonempty set of objects with sum and scalar
multiplication which assigns to each u, v εV , a sum u+v ε V ; and for
any k ε K and for any u ε V , the product ku ε V . Then V is a vector space
and the objects of V are vectors if V satisfies the following conditions:
For any u, v, w ε V , and for any k, k1, k2 ε k , we have:
i) u + (v+w) = (u+v) + w
ii) u+ v = v + u
iii) For any u ε V , there exists a 0 ε V such that 0 + u = u
iv) For any u ε V , there exists a − u ε V such that u + (-u) = 0
v) k(u+v) = ku+kv;
vi) (k1+k2)u = k1u + k2u;
vii) (k1k2)u = k1(k2u)
viii) 1u = u
Vector spaces and subspaces
Definition:
k ε ℜ => V is a real vector space
k ε ⊂ => V is a complex vector space
Typed by Ridwan Abrar, BUET EEE ‘16
,Examples:
1. All lines passing through the origin such as ax+by = 0 and ax+by+c
=0
3D = all planes passing through the origin
In case of 3D, a , b and c are direction ratios
2.
3. Set of all continuous functions
Linear Combination:
The vector u1, u2, …., un form a set S = { u1, u2, ….., un} for a vector space
V. Then any non-zero vector u ε V is a linear combination of the
vectors in S if u = c1u1 + c2u2 + c3u3 + …. + cnun
for any ci ε K , i = 1, n
Linear dependence:
The vectors u1, u2, …, un are linearly dependent if we can find scalars c1,
c2, ….. cn , not all of them zero such that c1u1+c2u2+...+cnun = 0
Example 1:
︿ ︿ ︿
Every vector in ℜ3 is a linear combination of i, j and k.
i = (1,0,0) = column of matrix = e1 =
j = (0,1,0) = e2
k = (0,0,1) = e3
Set u = (a,b,c) ε ℜ3
Then u = (a,b,c) = ae1 + be2 + ce3
= a(1,0,0) + b(0,1,0) + c(0,0,1)
Typed by Ridwan Abrar, BUET EEE ‘16
, Example 2
Examine whether the vectors w = (9,2,7) is a linear combination of u =
(1,2,-1) and v = (6,4,2).
Solution:
Take any scalars x,y such that
w = xu + yv
=> (9,2,7) = x(1,2,-1) + y(6,4,2) = (x,2x,-x) + (6y,4y,2y)
=> (9,2,7) = (x+6y, 2x+4y, -x+2y)
x + 6y = 9
2x+ 4y = 2
-x + 2y = 7
x = -3 and y = 2
(unique solution)
Non homogeneous system = None of the constants are zero
Consistent system = has a solution
w = -3u + 2v
Thus w is a linear combination of u and v.
Example 3
Examine whether w = (4,1,-8) is a linear combination of u, v.
w = xu + yv
=> (4,-1,8) = (x+6y, 2x+4y, -x+2y)
The system is:
Typed by Ridwan Abrar, BUET EEE ‘16
