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4 Vector Spaces 4.3 Linearly Independent Sets; Bases REVIEW Definition A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars, subject to the ten axioms: For all u, v, w in V and all scalars c and d, 1) u v V 2) u v v u 3)(u v) w u (v w) 4) There is a zero vector 0 such that u 0 u 5) For each u U , there is -u such that u (u ) 0 6) cu V 7)c(u v) cu cv 8)(c d)u cu du 9) c(du ) (cd)u 10)1u u Definition REVIEW A subspace of a vector space V is a subset H of V that satisfies a. The zero vector of V is in H. b. H is closed under vector addition. (For each u and v H , u v H) c. H is closed under multiplication by scalars. (For each u H and each scalar c, cu H ) REVIEW Theorem 1 If v1 , , v p are in a vector space V, then Span v1 , , v p is a subspace of V. REVIEW Definition The null space of an m n matrix A, written as Nul A, is the set of all solutions to Ax=0. Nul A x | Ax 0 and x R n Theorem 2 The null space of an m n matrix A is a subspace of n . REVIEW Definition The column space of an m n matrix A, written as Col A, is the set of all linear combinations of the columns of A. Col A Span{a1 ,, an } Note: Col A {b | Ax b for some x n } Theorem 3 m The column space of an m n matrix A is a subspace of . REVIEW Definition: A linear transformation T from a vector space V into a vector space W is a rule assigns to each vector x in V a unique vector T(x) in W, such that (i) T(u+v)=T(u)+T(v) for all u, v in the domain of T: (ii) T(cu)=cT(u) for all u and all scalars c. kernel of T x | T ( x) 0 and x V Range of T {b W | T ( x) b for some x V } If T(x)=Ax for some matrix A, Kernel of T = Nul A Range of T = Col A. 4.3 Linearly Independent Sets; Bases Purpose: To study the vectors that span a vector space (or a subspace) as efficiently as possible. Linear Independence v , v ,, v 1 2 p in V is linearly independent x1 v1 x2 v2 x p v p 0 has only the trivial solution. x1 v1 x2 v2 x p v p 0 x1 x2 x p 0 Tips to determine the linear dependence A set is linearly dependent, if it satisfies one of the following: 1. A set has two vectors and one is a multiple of the other. 2. A set has two or more vectors and one of the vectors is a linear combination of the others. 3. A set contains more vectors than the number of entries in each vector. 4. A set contains the zero vector. Theorem 4 An indexed set v1 , v2 , , v p of two or more vectors, with v1 0, is linearly dependent if and only if some v j ( j 1) is a linear combination of the preceding vectors, v1 , v2 , , v j 1. Example: p1 (t ) 3, p2 (t ) 2t , p3 (t ) t 1 Is p1 , p2 , p3 linearly dependent? Definition Let H be a subspace of a vector space V. An indexed set of vectors b1 ,, b p in V is a basis for H if i) is a linearly independent set, and ii) the subspace spanned by coincides with H; i.e. H Spanb1 , , b p Examples: 1. Let A be an invertible n n matrix. Then the columns of A form a basis for n. Why? 2. Let e1 , , en be the columns of the n n identity matrix I. Then, e1 ,, en is called the standard basis for n . 1 2 3 4. Let v1 2, v2 0, v3 0 3 1 1 . Determine if v1 , v2 , v3 is a basis for 3 . 1 2 2 0 5. Let v1 2, v2 0, v3 3 , v4 2. 3 1 1 1 Determine ifv1 , v2 , v3 , v4 is a basis for 3 . 4 2 6 6. Let v 6 , v 0 , v 9. 1 2 3 2 1 3 Determine if v1 , v2 , v3 is a basis for 3 . The Spanning Set Theorem Let S v1 , , v p be a set in V, and let H Spanv1 , , v p . a. If one of the vectors in S, say vk , is a linear combination of the remaining vectors in S, then the set formed from S by removing vk still spans H. b. If H 0, some subset of S is a basis for H. Example: Find a basis for Col A, where 1 0 A 0 0 3 0 0 0 4 2 0 0 3 4 1 2 0 0 0 1 Example: Find a basis for Col B, where 1 3 B 2 1 4 1 1 2 2 6 4 2 0 4 1 0 0 1 1 1 Theorem The pivot columns of a matrix A form a basis for Col A.