Linear Combination in Vector Sets Explained
Understanding linear combinations in vector sets through examples and visual representations. Explore how vectors can be combined using different coefficients to form a linear combination and determine if solutions exist.
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Presentation Transcript
Linear Combination Given a vector set ??,??, ,?? The linear combination of the vectors in the set ? = ?1??+ ?2??+ + ???? ?1,?2, ,??are scalars (coefficients of linear combination) 1 1,1 1+ 41 1 31 1 3, 3+ vector set: 1 1 =2 coefficients: 3,4,1 8 weighted sum
Column Aspect ?? ?? ?? ?1 ?2 ?? ? =?? ?? ?? coefficients ? = Vector set Linear Combination = ?1??+ ?2??+ + ????
System of Linear Equations v.s. Linear Combination Non empty solution set? Has solution or not? (A system of linear equations) Consistent? The Same question Column Aspect = ? Is ? the linear combination of columns of ?? = ?1??+ ?2??+ + ???? linear combination of columns of ?
Example 1 3?1+ 6?2= 3 2?1+ 4?2= 4 ?1 ?2 ? =3 6 4 ? =3 ? = 2 4 Has solution or not? Is ? the linear combination of columns of ?? 3 4 3 2,6 4
3?1+ 6?2= 3 2?1+ 4?2= 4 Example 1 Has solution or not? 3 2,6 Vector set: 4 Is 3 3 2,6 NO 4 a linear combination of ? 4 3 4 The linear combination is always on the dotted line. 6 4 3 2
Example 2 2?1+ 3?2= 4 3?1+ 1?2= 1 ?1 ?2 ? =2 3 1 4 ? = ? = 1 3 Has solution or not? Is ? the linear combination of columns of ?? 4 2 3,3 1 1
2?1+ 3?2= 4 3?1+ 1?2= 1 Example 2 Has solution or not? 2 3,3 4 1 2 3 1 23 1 3 1 4 1 (-1)2 3
Example 2 If u and v are any nonparallel vectors in R2, then every vector in R2 is a linear combination of u and v Nonparallel: u and v are nonzero vectors, and u cv. ?1?1 ?2?1 +?1?2 +?2?2 = ?1 = ?2 ? u and v are not parallel ? ? Has solution If u,v and w are any nonparallel vectors in R3, then every vector in R3 is a linear combination of u,v and w? NO
Example 3 2?1+ 6?2= 4 1?1+ 3?2= 2 ?1 ?2 ? =2 6 3 ? = 4 ? = 2 1 Has solution or not? Is ? the linear combination of columns of ?? 4 2 2 1,6 3
2?1+ 6?2= 4 1?1+ 3?2= 2 Example 3 Has solution or not? 2 1,6 Vector set: 3 2 1,6 Is 4 ? Yes 2 a linear combination of 3 6 3 2 1 4 2 u and v are not parallel Has solution
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
