4.1 VECTOR SPACES AND SUBSPACES


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1 4.1 VECTOR SPACES AND SUBSPACES What is a vector space? (pg 229) A vector space is a nonempty set, V, of vectors together with two operations; addition and scalar multiplication which satisfies the following axioms: 1.) u + v is in V 6.) c u is in V 2.) u + v = v + u 3.) ( u + v) + w = u + ( v + w) 7.) c (u + v) = cu + cv 8.) ( c + d)u = cu + du 4.) is in V 5.) u is in V and u + ( u) = 9.) c (du) = (cd) u 1.) 1 u = u These axioms must hold for all vectors u, v, and w in V and for all scalars c and d. What are P n, P, and M axb? What is a subspace? 91
2 What is the subspace test? Theorem 1 92
3 Example Determine if H { a + bt + ct + dt a + b c = } = is a subspace of P 3. 93
4 94 Example 2 Determine if + = + = + = z 4x 3y 2w & y x w z z y x w W is a subspace of 3 R. Determine if = = 2 b a d c b a H is a subspace of 4 R.
5 Example 3 Determine if a b c W = a + c = & b + d+ f = is a subspace of M 2x3. d e f Example 4 Let H be the set of all 2x2 matrices with determinant equal to 1. Determine if H is a subspace of M. 2x2 95
6 4.2 NULL SPACES, COLUMN SPACES, AND LINEAR TRANSFORMATIONS What is NulA? The null space of an mxn matrix A, written NulA, is the set of all solutions to the homogeneous equation A x = o. NulA = {x : x is in n R and A x = o } Theorem 2 The null space of an mxn matrix A, NulA, is a subspace of n R. What is ColA? The column space of an mxn matrix A, written ColA, is the set of all linear combinations of the columns of A. If A = [a 1 a 2 K a n ], then ColA = Span{a1, a 2, K, a n }. Theorem 3 The column space of an mxn matrix A, ColA, is a subspace of m R. The column space of an mxn matrix, A is all of m R. m R if and only if the equation A x = b has a solution for each b in What is a linear transformation? A linear transformation T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T ( x) in W where the following conditions hold: 1.) T ( ) = (This condition is redundant because c in condition 3 could be. However, it is easy to check so is often looked at first.) 2.) T (u + v) = T(u) + T(v) for all u and v in V. 3.) T (cu) = ct(u) for all u in V and all scalars c. 96
7 CONTRAST BETWEEN NulA AND ColA FOR AN mxn MATRIX A NulA ColA 1.) NulA is a subspace of n R. 2.) NulA is implicitly defined; that is, you are given a condition ( A x = ) that each x in NulA must satisfy. 3.) It takes time to find vectors in NulA. Row operations on [A ] are required. 4.) There is no obvious relation between NulA and the entries in A. 5.) A typical vector v in Nul A has the property 1.) ColA is a subspace of m R. 2.) ColA is explicitly defined; that is, you are told how to build vectors in ColA. 3.) It is easy to find vectors in ColA. The columns of A are displayed; others are formed from them. 4.) There is an obvious relation between ColA and the entries in A, since each column of A is in ColA. 5.) A typical vector v in ColA has the property that A v =. that the equation A x = v is consistent. 6.) Given a specific vector v it is easy to tell if v 6.) Given a specific vector v, it may take time to is in NulA, just compute A v. 7.) NulA = {} if and only if the equation A x = has only the trivial solution. 8.) NulA = {} if and only if the linear transformation x a Ax is onetoone. 7.) 8.) tell if v is in ColA. Row operations on [A v ] are required. m ColA = R if and only if the equation A x = b has a solution for every b in m ColA = R if and only if the linear transformation x a Ax maps n R onto m R. m R. 97
8 What is the kernel of a linear transformation? 98
9 Example 1 a b c a + b 2c Determine if T : M 2 x3 M2x2 defined by T = is a linear d e f e + 3f b + d transformation. Find the kernel of T. 99
10 4.3 LINEARLY INDEPENDENT SETS; BASES Linear Independence vs. Linear Dependence The set of vectors { v1, v2, K, vp} in V is : 1.) linearly independent if the vector equation c v + c v + K + c v has only the trivial solution ( c = c = K = c ). 1 2 p = p p = 2.) linearly dependent if the vector equation c v + c v + K + c v has a non trivial solution p p = Theorem 4 An set { v1, v2, K, vp} of two or more vectors in V, with v 1, is linearly dependent if and only if some j > 1) is a linear combination of the preceding vectors, v1, v2, K, v j 1. v j (with What is a basis? Let H be a subspace of a vector space V. A set of vectors B = b1, b2, K, bp in V is a basis for H if both of the following conditions hold: 1.) B is a linearly independent set 2.) The subspace spanned by B is H; that is, H = Span b1, b2, K, bp. What is the standard basis of P n, P, and M axb? 1
11 Example 1 Let T : P 6 P3 be a linear transformation defined by T at + bt + ct + dt + et + ft + g = 2a 6b + 4c t for the kernel of T. 3 2 ( ) ( ) + ( f g) t + ( a 3b) t c. Find a basis 11
12 Theorem 5: The Spanning Set Theorem Theorem 6 The pivot columns of a matrix A form a basis for ColA. 12
13 Example 2 a + 2b 3c 3a + 6b 5c + 4d 3a + 5b 11c + 6d Let H = a, b, c, d R be a b + 6c 2d a + 3b + 3c 2d 2a + 9b 2c + 2d subspace of M 2x3. Find a basis for H. 13
14 4.4: THE DIMENSION OF A VECTOR SPACE Theorem 7 Theorem 8 What is the dimension of a vector space? What is a finite/infinite dimensional vector space? 14
15 Example 1 True or False 1.) If there exists a linearly dependent set { v1, v 2, K, v p } in V then dim( V) p. 2.) If every set of p elements in V fails to span V then dim( V) > p. 3.) If p 2 and dim( V) = p, then every set of p1 nonzero vectors is linearly independent. 15
16 Theorem 9 Theorem 1 (The Basis Theorem) Let V be a pdimensional vector space, p 1, and let S = {v, v, v } be a set of p vectors in V. a.) If S is linearly independent, then it is a basis for V. b.) If S spans V, then it is a basis for V. 1 2, K p Dimensions of NulA & ColA 16
17 17 Example 2 Let = + + = + + = + = f d c & f, d 2c a f, d 2c b a f c e b d a H. Find the dimension of H.
18 Example 3 Which of the following are bases for P 2? 1.) t + t + 2, 2t + 2t + 3, 4t 1 18
19 2.) t t 1, 2t + 3t 2 19
20 3.) 3t t + 1, t + t + 1, t
21 4.5: RANK What is the row space of a matrix? Theorem 11 What is the rank of a matrix? The rank of a matrix A, written rank(a), is the dimension of the column space of A. Theorem 12 (The Rank Theorem) 111
22 112 Example 1 The matrices below are row equivalent. = A & = B 1.) Find ColA, NulA, and RowA.
23 2.) Find a basis for ColA and NulA. 3.) Find two different bases for RowA. 4.) What is Rank(A) and dim(nula)? 113
24 Example 2 1.) If A is a 3x4 matrix, what is the largest/smallest possible value for ranka and dim(nula)? 2.) If A is a 4x6 matrix, are the columns of A linearly independent or linearly dependent? 114
25 3.) If A is a 5x3 matrix, are the rows of A linearly independent or linearly dependent? 4.) If A is a 12x7 matrix and dim ( NulA) 4 =, what is the dimension of RowA? 115
26 The Invertible Matrix Theorem (pg 163, 19, & 297) Let A be a square nxn matrix. Then the following statements are equivalent. That is, for a given matrix A, the statements are either all true or all false. 1.) A is an invertible matrix. 2.) A is row equivalent to the nxn identity matrix, I n. 3.) A has n pivot positions. 4.) The equation A x = has only the trivial solution. 5.) The columns of A form a linearly independent set. 6.) The linear transformation x a Ax is onetoone. 7.) The equation A x = b has at least 1 solution for each b in 8.) The columns of A span n R. 9.) The linear transformation x a Ax maps n R onto n R. 1.) There is an nxn matrix C such that AC = CA = In. 11.) T A is an invertible matrix. 12.) det A 13.) The columns of A form a basis for 14.) n ColA = R 15.) dim ( ColA) = n 16.) rank ( A) = n 17.) NulA = {} 18.) dim ( NulA) = n R. 19.) The number is not an eigenvalue of A. n R. 116
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