This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #5 (due Thursday, October 5) All problems are from Leon’s book (8th edition). Section 3.3: 2b, 2c, 2e, 8c, 9d, 14 Section 3.4: 8c, 10, 14c, 15a MATH 304 Linear Algebra Lecture 11: Basis and dimension. Linear independence Definition. Let V be a vector space. Vectors v 1 , v 2 , . . . , v k ∈ V are called linearly dependent if they satisfy a relation r 1 v 1 + r 2 v 2 + ··· + r k v k = , where the coefficients r 1 , . . . , r k ∈ R are not all equal to zero. Otherwise the vectors v 1 , v 2 , . . . , v k are called linearly independent . That is, if r 1 v 1 + r 2 v 2 + ··· + r k v k = = ⇒ r 1 = ··· = r k = 0. An infinite set S ⊂ V is linearly dependent if there are some linearly dependent vectors v 1 , . . . , v k ∈ S . Otherwise S is linearly independent . Remark. If a set S (finite or infinite) is linearly independent then any subset of S is also linearly independent. Theorem Vectors v 1 , . . . , v k ∈ V are linearly dependent if and only if one of them is a linear combination of the other k − 1 vectors. Examples of linear independence. • Vectors e 1 = (1 , , 0), e 2 = (0 , 1 , 0), and e 3 = (0 , , 1) in R 3 . • Matrices E 11 = parenleftbigg 1 0 0 0 parenrightbigg , E 12 = parenleftbigg 0 1 0 0 parenrightbigg , E 21 = parenleftbigg 0 0 1 0 parenrightbigg , and E 22 = parenleftbigg 0 0 0 1 parenrightbigg . • Polynomials 1 , x , x 2 , . . . , x n , . . . Spanning set Let S be a subset of a vector space V . Definition. The span of the set S is the smallest subspace W ⊂ V that contains S . If S is not empty then W = Span ( S ) consists of all linear combinations r 1 v 1 + r 2 v 2 + ··· + r k v k such that v 1 , . . . , v k ∈ S and r 1 , . . . , r k ∈ R . We say that the set S spans the subspace W or that S is a spanning set for W . Remark. If S 1 is a spanning set for a vector space V and S 1 ⊂ S 2 ⊂ V , then S 2 is also a spanning set for V . Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis . Suppose that a set S ⊂ V is a basis for V . “Spanning set” means that any vector v ∈ V can be represented as a linear combination v = r 1 v 1 + r 2 v 2 + ··· + r k v k , where v 1 , . . . , v k are distinct vectors from S and r 1 , . . . , r k ∈ R . “Linearly independent” implies that the above representation is unique: v = r 1 v 1 + r 2 v 2 + ··· + r k v k = r ′ 1 v 1 + r ′ 2 v 2 + ··· + r ′ k v k = ⇒ ( r 1 − r ′ 1 ) v 1 + ( r 2 − r ′ 2 ) v 2 + ··· + ( r k − r ′ k ) v k = = ⇒ r 1 − r ′ 1 = r 2 − r ′ 2 = . . . = r k − r ′ k = 0 Examples. • Standard basis for R n : e 1 = (1 , , , . . ., , 0), e 2 = (0 , 1 , , . . ., , 0),. . . , e n = (0 , , , . . ., , 1)....
View
Full
Document
This note was uploaded on 11/08/2011 for the course MATH 304 taught by Professor Hobbs during the Spring '08 term at Texas A&M.
 Spring '08
 HOBBS
 Linear Algebra, Algebra

Click to edit the document details