Understanding Vector Dependency and Independence
This content discusses the concepts of linear dependency and independence of vector sets, providing definitions and criteria for determining whether a set of vectors is dependent or independent. It includes examples and explanations for better comprehension of the topic.
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() Dependent and Independent ( )
Definition A set of n vectors ?1,?2, ,?? is linear dependent If there exist scalars ?1,?2, ,??, not all zero, such that ?1?1+ ?2?2+ + ????= ? 2 2 Obtain many Find one 2 A set of n vectors ?1,?2, ,?? is linear independent ?1?1+ ?2?2+ + ????= ? Only if ?1= ?2= = ??= 0 unique
A set of n vectors ?1,?2,,?? is linear dependent If there exist scalars ?1,?2, ,??, not all zero, such that ?1?1+ ?2?2+ + ????= ? A set of n vectors ?1,?2, ,?? is linear independent ?1?1+ ?2?2+ + ????= ? Only if ?1= ?2= = ??= 0 4 12 6 10 30 15 Dependent or Independent? , dependent 4 12 6 10 30 15 ?1 5 + ?2 -2 = ?
A set of n vectors ?1,?2,,?? is linear dependent If there exist scalars ?1,?2, ,??, not all zero, such that ?1?1+ ?2?2+ + ????= ? A set of n vectors ?1,?2, ,?? is linear independent ?1?1+ ?2?2+ + ????= ? Only if ?1= ?2= = ??= 0 7 6 3 3 1 8 3 Dependent or Independent? , , 11 6 dependent 7 11 6 6 3 3 1 8 3 ?1 1 + ?2 + ?3 -1 = ? 1
A set of n vectors ?1,?2,,?? is linear dependent If there exist scalars ?1,?2, ,??, not all zero, such that ?1?1+ ?2?2+ + ????= ? A set of n vectors ?1,?2, ,?? is linear independent ?1?1+ ?2?2+ + ????= ? Only if ?1= ?2= = ??= 0 3 0 0 0 2 5 1 Dependent or Independent? , , 1 7 dependent 3 0 0 0 2 5 1 ?1 0 + ?2 Any + ?3 = ? 1 7 0 Any set contains zero vector would be linear dependent
Linear Dependent (for ? 2) Given a vector set, {a1, a2, , an}, there exists scalars x1, x2, , xn, that are not all zero, such that x1a1 + x2a2+ + xnan= 0. ?1??+ ?2?? + ????+ + ????= ? 0 ?? 0 ?1??+ ?2?? + ????= ???? ?? ?1 ?? ?2 ?? ?? ?? ?? ??= ?? Given a vector set, {a1, a2, , an}, if there exists any ai that is a linear combination of other vectors
Linear Dependent (for ? 2) Given a vector set, {a1, a2, , an}, there exists scalars x1, x2, , xn, that are not all zero, such that x1a1 + x2a2+ + xnan= 0. ??= ?1??+ ?2?? + ???? ?1?? ?2?? + ?? ????= ? 0 ?? 0 Given a vector set, {a1, a2, , an}, if there exists any ai that is a linear combination of other vectors
Vector Set Linear Dependent = Given a vector set, {a1, a2, , an}, there exists scalars x1, x2, , xn, that are not all zero, such that x1a1 + x2a2+ + xnan= 0. (for ? 2) ai Given a vector set, {a1, a2, , an}, if there exists any ai that is a linear combination of other vectors Vector Set Linear Independent =
Summary ? ?? ? ?? ?:? ? Is ? in the span of the columns of ?? Is ? a linear combination of columns of ?? YES NO The columns of ? are independent. The columns of ? are dependent. No solution Rank A = n Rank A < n Nullity A = 0 Unique solution Nullity A > 0 Infinite solution
Dependent: Once we have solution, we have infinite. Intuition Intuitive link between dependence and the number of solutions 6 1 7 3 8 11 3 3 6 ?3 12 6 3 3 3 6 6 3 3 3 12 dependent ?1 ?2 7 6 3 3 1 8 3 14 22 1 + 1 = 11 6 = ?1 ?2 ?3 7 1 1 1 1 8 14 22 12 = 1 + 1 + 1 = 11 ?1 ?2 ?3 1 8 14 22 2 2 0 Infinite Solution 2 + 2 = =
Proof Columns of A are dependent If Ax=b have solution, it will have Infinite solutions If Ax=b have Infinitesolutions Columns of A are dependent
Homogeneous linear equations Proof ?? = ? (always having ? = ? as solution) Columns of A are dependent If Ax=0 have solution, it will have Infinite solutions ?? ? = ?1 ?? = ? ?1?1+ ?2?2+ + ????= ? there exist scalars ?1,?2, ,??, not all zero ?2 dependent set If Ax=0 have Infinitesolutions Columns of A are dependent ?? = ? have non zero solutions ?1?1+ ?2?2+ + ????= ? ?1,?2, ,??, not all zero
Proof Ax=0 have infinite solutions Columns of A are dependent If Ax=b have solution, it will have Infinite solutions We can find non-zero solution u such that ?? = ? ? ? + ? = b ? + ? is another solution different to v There exists v such that ?? = ? If Ax=b have Infinitesolutions Columns of A are dependent ?? = b ?? = b Ax=0 have infinite solutions ? ? ? ? ? = ? Non-zero
