Worksheet for Lecture 15 (due October 23) Section 4.3 Linearly Independent Sets; Bases

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1 Worksheet for Lecture 5 (due October 23) Name: Section 4.3 Linearly Independent Sets; Bases Definition An indexed set {v,..., v n } in a vector space V is linearly dependent if there is a linear relation among them: c v + + c n v n =, with c,..., c n not all zero. Otherwise, say that {v,..., v n } is linearly independent Theorem An indexed set {v,..., v n } of two or more vectors, with v, is linearly dependent if and only if some v j (with j > ) is a linear combination of the preceding vectors v,..., v j. Example For p (t) =, p 2 (t) = t, and p 3 (t) = 4 t. Example 2 In the vector space of functions on R, the set {sin t, cos t} the set {sin 2t, sin t cos t} Big Concept Let H be a subspace of a vector space V. B = {v,..., v n } in V is a basis of H if () B is a linearly independent set, and (2) we have H = Span{v,..., v n } Example R n has a basis: e =., e 2 =.,..., e n =. {e,..., e n } is a standard basis of R n. An indexed set of vectors Example 2 In general, vectors v,..., v n R n forms a basis if and only if the matrix A = v v 2 v n is invertible

2 2 Example 3 Let P n denote the vector space of polynomials of degree n. Let S =, t, t 2,..., t n }. Then S forms a basis of S. Spanning set theorem Let S = {v,..., v n } be a set in V, and let H = Span{v,..., v n }. (a) If one of the vectors in S, say v k, is a linear combination of the remaining vectors in S, then the set formed from S by removing v k still spans H. (b) If H {}, some subset of S is a basis of H. Illustration by example Find a basis for Col(A) with A = 4 2 v v 2 v 3 v 4 v 5 = Theorem Say A be an echelon form as above, and B is a matrix that is row equivalent to A. Say B = ( w w 2 w 3 w 4, w 5 ). Then Col(A) = Span{v, v 3, v 5 } Col(B) = Span{w, w 3, w 5 } In general, the column space of B is spanned by the columns of B indexed by the pivot columns of A

3 Example Find a basis of Col(A) for A = A good philosophy to remember: A basis of H is a linearly independent subset of H that is as large as possible; and it is at the same time a spanning set of H that is as small as possible. Section 4.4 Coordinate System Theorem (The Unique Representation Theorem) Let B = {b,..., b n } be a basis for a vector space V. Then for each x in V, there exists a unique set of scalars c,..., c n such that x = c b + c 2 b c n b n. Example For V = R 4 and the standard basis e =, e 2 =, e 3 =, e 4 =. a We can write each vector b c uniquely as ae + be 2 + ce 3 + de 4. d Definition Suppose that B = {b,..., b n } is a basis for V and x is in V. The coordinates of x relative to the basis B (or the B-coordinate of x) are the weights c,..., c n such that x = c b + + c n b n. We write [ x ] B = c. and call it the coordinate vector of x (relative to B). Then the map x [ x ] is the B coordinate mapping V R n. c n

4 4 { Example Consider a basis B = b =, b 2 = 2, b 3 = 4 } of R 3. What s 3 9 the B-coordinate of a vector x = 5? 7 Coordinate mapping Theorem Let B = {b,..., b n } be a basis for a vector space V. Then the coordinate mapping x [ x ] B is a one-to-one linear transformation from V onto Rn. (We say that V is isomorphic to R n.) Example Let B = {, t, t 2, t 3 } be the standard basis of the space P 3 of polynomials of degree 3. Then the coordinate mapping is given by Example 2 With the same notation as above, B = { + t, t, t 2, t 3 + t 2 } is also a basis of P 3, then the coordinate mapping with respect to B is given by Upshot: The vector space P 3 is isomorphic to R 4. But there are many ways to write this linear isomorphism, and this depends on a choice of basis. It is an important concept in linear algebra to allow working with different basis of P 3, and therefore different ways to realize P 3 as a more concrete vector space R 4.

5 5 True/False Questions () If H = Span{b,..., b n }, then {b,..., b n } is a basis for H. (2) The columns of an invertible matrix form a basis for R n. (3) A linearly independent set in a subspace H is a basis for H. (4) If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V. (5) A basis is a linearly independent set that is as large as possible. (6) If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col(A). (7) If x is in V and if B contains n vectors, then the B-coordinate vector of x is in R n. (8) If B is the standard basis for R n, then the B-coordinate vector of an x R n is x itself. Exercise Consider a basis B = { b =, b 2 = 3 3 4, b 3 = } of R 3. 4 What s the B-coordinate of a vector x = 8 9? 6

6 6 Exercise 2 The set B = { + t 2, t + t 2, + 2t + t 2 } is a basis for P 2. Find the coordinate vector of p(t) = + 4t + 7t 2 relative to B. Exercise 3 Find a basis for Col(A) and for Nul(A) with 3 2 A =