Lecture 07

5
Examples and Definition
Example. Let
S {( x1 , x2 , x3 )T | x1 x2 } .
S
is
nonempty
since
x (1,1,0)T S . To show S is a subspace of R 3 , we need to verify that the
(a, a, b)T (c, c, d )T (a c, a c, b d )T S
Since S is nonempty and satisfies the two closure conditions, it follows that S is a subspace of R 3 .
is called the span of v1 , v 2 ,
Span(v1 , v 2 ,
, vn ) . , v n are elements of a vector space V , then Span(v1 , v 2 , , vn )
Theorem. If v1 , v 2 , is a subspace of V mp; Linear Independent
1
Examples and Definition
Given a vector space, it is often possible to form another vector space by taking a subset S of V and using the operations of V . Since V is a vector space, the operations of addition and scalar multiplication always produce another vector in
a b A b c Since the (2,1) entry of A is the negative of the (1, 2) entry, A S .
(ii) If A, B S , then they must be of the form
x1 x3 x4 x 2 2 x 3 x4
11
The Nullspace of a Matrix
Example. Determine N ( A) if
1 1 1 0 A 2 1 0 1
Solution. (continue) Thus, if we set x3 and x4 , then
9
The Nullspace of a Matrix
Let A be an m n matrix. Let N ( A) denote the set of all solutions to the homogeneous system Ax 0 . Thus
N ( A) {x Rn | Ax 0} Since 0 N ( A) , so N ( A) is nonempty. If x N ( A) and is a scalar, then A( x) Ax 0 0
14
Span and Spanning Sets
Definition. The set {v1 , v 2 ,
, v n } is a spanning set for V if and only if every , vn .
vector in V can be written as a linear combination of v1 , v 2 , Example. Which of the following are spanning sets for R 3 ? A. {e1 ,e2 ,e3 ,(1,2,3)T } B. {(1,1,1)T ,(1,1,0)T ,(1,0,0)T } C. {(1,0,1)T ,(0,1,0)T } D. {(1,2,4)T ,(2,1,3)T ,(4, 1,1)T }
2
Examples and Definition
c x1 2 Example. Let S x2 2 x1 . S is a subset of R . If is any 2c x2
c c element of S and is any scalar, then which is an element 2 c 2 c a b of S . If and are any two elements of S , their sum 2a 2b a b a b 2a 2b 2(a b ) is also an element of S . Then, by the definition of vector space, the mathematical system consisting of the set S , together with the
1 1 2 2 1 x 1 0 0 1 is a solution to Ax 0 . The vector space N ( A) consists of all vectors of the
two closure properties hold. (i) If x (a, a, b)T S , then
x ( a, a, b)T
(ii) If (a, a, b)T and (c, c, d )T are arbitrary elements of S , then
8
Examples and Definition
a b d A and B e b c e f
It follows that
a d b e A B (b e ) c f
Hence A B S .
10
The Nullspace of a Matrix
Example. Determine N ( A) if
1 1 1 0 A 2 1 0 1 Solution. Using Gauss-Jordan reduction to solve Ax 0 , we obtain
form
12
The Nullspace of a Matrix
Example. Determine N ( A) if
1 1 1 0 A 2 1 0 1
Solution. (continue)
1 1 2 1 1 0 0 1
6
Examples and Definition
Example. Let S { A R22 | a12 a21 }. The set S forms a subspace of R 2 2 , since (i) If A S , then A must be of the form a b A b c and hence
and hence x N ( A) . If x and y are elements of N ( A) , then A(x y) Ax Ay 0 0 0 Therefore, x y N ( A) . It follows then that N ( A) is a subspace of R n . The set of all solutions to the homogeneous system Ax 0 forms a subspace of R n . The subspace N ( A) is called the nullspace of A .
1 1 1 0 0 1 1 1 0 0 2 1 0 1 0 0 1 2 1 0 1 0 1 1 0 1 0 1 1 0 0 1 2 1 0 0 1 2 1 0 The reduced row echelon form involves two free variables, x3 and x4 .
V . For a new system using a subset S of V as its universal set to be a vector
space, the set S must be closed under the operations of addition and scalar multiplication. Then S also can be thought as a vector space. The following examples show that how these work.
n v n , where 1 ,
, n are scalars, is called a linear
, vn , v n will be denoted by
, v n . The set of all linear combinations of v1 , v 2 , , v n . The span of v1 , v 2 ,
where and are scalars. Finish.
13
Span and Spanning Sets
Definition. Let v1 , v 2 , form 1 v1 2 v 2 combination of v1 , v 2 ,
, v n be vectors in a vector space V . A some of the
4
Examples and Definition