Linear Algebra: UNIT I: Vector Spaces
Definition of Linear Dependence and Linear Independence
Vector Spaces: Definition of Linear Dependence and Linear Independence
LINEAR DEPENDENCE AND
LINEAR INDEPENDENCE
Definition: Linear Dependences
A
subset S of a vector space V is called linearly dependent if there exist a
finite number of distinct vectors u1, u2, u3….
un in S and scalars a1, a2, a3 ... an not
all zero, such that
а1u1 + а2u2
+ а3u3 +.. .. + аnun = 0.
In
this case we say that the vectors of S are linearly dependent.
For
any vectors u1, u2, u3... un we
have а1u1 + а2u2 + а3u3
+.. .. + аnun = 0. If а1= а2= а3
=…. аn=0. We call this the trivial representation of 0 as a
linear combination of u1, u2, u3 ... un.
Thus for a set to be linearly dependent, there must exist a non−trivial
representation of 0 as a linear combination of vectors in the set.
Consequently,
any subset of a vector space that contains the zero vector is linearly
dependent, because 0= 1.0 is a nontrivial representation of 0 as a linear
combination of vectors in the set.
Definition: Linear Independence
A
subset S of a vector space that is not linear dependent is called linearly
independent. We also say that the vectors of S are linearly independent.
The
following facts about linearly independent sets are true in any vector space.
(i) The empty set is linearly independent, for
linearly dependent, sets must be non−empty.
(ii)
A set consisting of a single non−zero vector is linearly independent. If {u} is
linearly dependent, then au= 0 for
some non−zero scalar a.
u = a‒1(au)
= a‒1 0 = 0.
(iii)
A set is linearly independent if and only if the only representations of 0 as
linear combinations of its vectors are trivial representations.
Linear Algebra: UNIT I: Vector Spaces : Tag: maths, mathematics : - Definition of Linear Dependence and Linear Independence
Linear Algebra: UNIT I: Vector Spaces
Under Subject
Linear Algebra
MA25C02 2nd Semester | 2025 Regulation
Related Subjects
English Essentials II
EN25C02 2nd Semester | 2025 Regulation | 2nd Semester 2025 Regulation
Tamils and Technology தமிழர்களும் தொழில்நுட்பமும்
UC25H02 2nd Semester | 2025 Regulation | 2nd Semester 2025 Regulation
Linear Algebra
MA25C02 2nd Semester | 2025 Regulation
Transforms and its Applications
MA25C03 2nd Semester EEE Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Applied Physics (CE) II
PH25C02 2nd Semester Civil, Agri Depts | 2025 Regulation | 2nd Semester 2025 Regulation
Applied Physics (CSIE) II
PH25C03 2nd Semester AIDS, CSE, IT, CSE(CY) Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Applied Physics (EE) II
PH25C04 2nd Semester EEE Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Applied Physics (ME) II
PH25C05 2nd Semester Mechanical Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Applied Chemistry (CE) II
CY25C02 2nd Semester Civil Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Applied Chemistry (ME) II
CY25C03 2nd Semester Mechanical Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Electron Devices
EC25C01 2nd Semester ECE Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Digital Principles and Computer Organization
CS25C06 2nd Semester AIDS, CSE, IT, CSE(CY) Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Basic Electrical and Electronics Engineering
EE25C01 2nd Semester | 2025 Regulation | 2nd Semester 2025 Regulation
Basic Civil and Mechanical Engineering
GE25C01 2nd Semester EEE Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Data Structures using CPlusPlus
CS25C05 2nd Semester ECE Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Engineering Drawing
ME25C01 EEE, Mech, Agri, EEE Depts | 2025 Regulation | 2nd Semester 2025 Regulation
Data Structures and Algorithms
CS25C04 2nd Semester EEE Dept | 2025 Regulation
Circuits and Network Analysis
EC25C02 2nd Semester ECE Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Engineering Mechanics
ME25C02 2nd Semester Mech, Civil, Agri Depts | 2025 Regulation | 2nd Semester 2025 Regulation
Object Oriented Programming (OOPs)
CS25C07 2nd Semester CSE, CSE(CY) Depts | 2025 Regulation | 2nd Semester 2025 Regulation