3.2 Subspace. Definition: If S is a non-empty subset of a vector space V, and S satisfies the following conditions: (i).

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1 . ubspace Given a vector spacev, it is possible to form another vector space by taking a subset of V and using the same operations (addition and multiplication) of V. For a set to be a vector space, it must satisfy addition and multiplication closure conditions on top of the other 8 conditions on addition and multiplication which are naturally satisfied since is a subset of V, so we just need to check addition and multiplication closure conditions. If the subset is a vector space, we call vector space is a subspace of V. Definition: If is a non-empty subset of a vector space V, and satisfies the following conditions: (i). x, whenever x for any scalar (ii). x y, whenever x, y Then we call is a subspace of V. Remark: Every vector space V, has at least two subspaces. Namely V itself and 0 (the zero space). These two subspaces are called trivial subspaces. Example 1: Determine if the given set is a subspace of the given vector space. (1). Let x, x. Is a subspace of vector space 1 1 Let x ( x, x ), y ( y, y ), is a scalar. x ( x, x ) 1 1 x y ( x y, x y ) ( x y, ( x y )) is a subspace (). Let x, x x 0, x 0. Is a subspace of vector space 1 1 x (1,), y ( 1,1) x y (0,) is not a subspace. (). Let x, x x 0. Is a subspace of vector space 1 1 x(1,), x (, 4) is not a subspace.

2 (). Let x 1,0. Is a subspace of vector space Let x ( x1,0), y ( y1,0), is a scalar. x y ( x y,0) 1 1 x ( x,0) 1 is a subspace. (). Let x 1,1. Is a subspace of vector space x(,1), but x(4,) is not a subspace. Exercise: Determine if the given set is a subspace of the given vector space. (1). Let be the set of diagonal matrices. Is a subspace of vector space M? (yes) (). Let be the set of matrices of the form (). Let be the set of matrices of the form a 0 a 1 b c b c. Is a subspace of vector space M. Is a subspace of vector space M? (yes)? (No) Exercise: Determine if the given set is a subspace of the given vector space. (1). Let C[ a, b] be the set of all continuous functions on the interval [ ab., ] Is this a subspace of F[ a, b ], the set of all real valued functions on the interval [ ab?, ] (yes) (). Let Pn be the set of all polynomials of degree n or less. Is this a subspace of F[ a, b ]? (yes) (). Let be the set of all polynomials of degree exactly. Is this a subspace of F[ a, b ]? (no) f ( x) x, g( x) x x, so it is not a subspace ( f g)( x) x (4). Let be the set of all functions such f () 1. Is this a subspace of F[ a, b ]? (no)

3 f ( x) x 1, g( x) ( f g)( x) x 1 x 1 ( f g)() 1 so it is not a subspace (5). Let be the set of all functions such f () 0. Is this a subspace of F[ a, b ]? (yes) Exercise: Let A R. be a particular vector in R. Determine whether the following are subspaces of (a). { B R AB BA} Let B, C, is a scalar. B, C BA AB, CA AC ( B C) A BA CA AB AC A( B C) B C ( B) A BA AB A( B) B o it is a subspace (b). { B R AB BA} Let B. 0A 0 A0 0 0B 0 o it is not a subspace (c). { B R AB 0} (yes). Null pace of a matrix. Definition: Let A be an m nmatrix. The null space N( Aof ) A is the set N( A) { x R Ax 0} Property: N( A ) is a subspace. Let xa, y A, be a scalar. n

4 x A, y A Ax Ay 0 A( x y) Ax Ay x y A A( x) Ax 0 0 x A o N( Ais ) a subspace. Example : Determine N( A ) if 1 0 A R1 R R R R A R R R R1 R o, we have x1 x x4 x x x 4 Let x, x4, we have x1 x x x4 x1 1 x x x 1 0 x4 0 1 N( Acontains ) all vectors of the form: where, are any scalars.

5 N( Ais ) a linear combination of vector definitions below. 1, or the span of 1, We give these Definition: Let v 1, v,, vn be vectors in a vector space V, 1,,, n be scalars, v v v 1 1 n n is called the linear combinations of vectors v 1, v,, v n, while W { v v v,,, are scalars} 1 1 n n 1 n be called the span of v 1, v,, v n, denoted by span( v1, v,, v n), or Example: Let e 1 0, e 1, e span( e ) is a line ( x-axis) 1 span( e, e ) is a plane ( xy-plane) 1 span( e, e, e ) R 1 W span( v, v,, v n )., what are span( e1 ), span( e1, e ), span( e1, e, e ) in Definition: The set { v1, v,, v n } is called a spanning set for vector space V if every vector in V can be written as a linear combination of v1, v,, v n. Example: Let E11 1 0, E , E1, E , is { E11, E1, E1, E } a spanning set for vector space: all matrices? a b For any matrix, ae be ce de c d ,so it is a spannng set. Exercise: Let E11, E1, E1, E, v , is { E11, E1, E1, E } a spanning set for vector space: all matrices?. a b For any matrix, ae11 be1 ce1 de 0 v,so it is a spannng set. c d

6 1 0 1 Exercise: Let v 1 0, v 1, v, { v1, v, v} is a spanning set for vector space For any vector u1 u u u whether we can find scalars 1,, such that 1 1 v v v u or u u u 0 1 u u u (What we need to do is to determine if this system has at least one solution for every possible vector u. If the coefficient matrix determinant is non-zero, then the system has a unique solution, so we just need to compute the determinant of coefficient matrix. ) Thus, there are unique scalars 1,, such that 1v1 v v u. Thus, { v1, v, v} is a spanning set for vector space R. Exercise: Let v1 x 1, v x x, v x 1 { v1, v, v} is a spanning set for vector space, the polynomial with degree or less? P { ax bx c a, b, c are scalars} For any vector P( x) ax bx c P whether we can find scalars 1,, such that 1v1 v v P() x or 1( x 1) ( x x) ( x 1) ax bx c. 1( 1) ( ) ( 1) ( ) ( 1 ) 1 x x x x ax bx c x x ax bx c a a b 1 0 b (. 1) c c Thus, there are unique solution to equation (.-1). Thus, there are unique scalars 1,, such that 1v1 v v P() x. Thus, { v1, v, v} is a spanning set for vector space P. HW: 1,10(d), 14(c).