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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #4 (due Thursday, September 29) All problems are from Leon’s book (8th edition). Section 3.1: 9a, 9b, 11 Section 3.2: 1b, 1c, 11b, 11c, 13a, 13b, 20 MATH 304 Linear Algebra Part II ( ≈ 4.5 weeks): Abstract linear algebra • Vector spaces • Linear independence • Basis and dimension • Coordinates, change of basis • Linear transformations Leon’s book : Chapters 3–4 MATH 304 Linear Algebra Lecture 8: Vector spaces. Subspaces. Linear operations on vectors Let x = ( x 1 , x 2 , . . . , x n ) and y = ( y 1 , y 2 , . . . , y n ) be ndimensional vectors, and r ∈ R be a scalar. Vector sum: x + y = ( x 1 + y 1 , x 2 + y 2 , . . . , x n + y n ) Scalar multiple: r x = ( rx 1 , rx 2 , . . . , rx n ) Zero vector: = (0 , , . . ., 0) Negative of a vector: − y = ( − y 1 , − y 2 , . . . , − y n ) Vector difference: x − y = x + ( − y ) = ( x 1 − y 1 , x 2 − y 2 , . . . , x n − y n ) Properties of linear operations x + y = y + x ( x + y ) + z = x + ( y + z ) x + = + x = x x + ( − x ) = ( − x ) + x = r ( x + y ) = r x + r y ( r + s ) x = r x + s x ( rs ) x = r ( s x ) 1 x = x x = ( − 1) x = − x Linear operations on matrices Let A = ( a ij ) and B = ( b ij ) be m × n matrices, and r ∈ R be a scalar. Matrix sum: A + B = ( a ij + b ij ) 1 ≤ i ≤ m , 1 ≤ j ≤ n Scalar multiple: rA = ( ra ij ) 1 ≤ i ≤ m , 1 ≤ j ≤ n Zero matrix O: all entries are zeros Negative of a matrix: − A = ( − a ij ) 1 ≤ i ≤ m , 1 ≤ j ≤ n Matrix difference : A − B = ( a ij − b ij ) 1 ≤ i ≤ m , 1 ≤ j ≤ n As far as the linear operations are concerned, the m × n matrices have the same properties as mndimensional vectors. Abstract vector space: informal description Vector space = linear space = a set V of objects (called vectors ) that can be added and scaled. That is, for any u , v ∈ V and r ∈ R expressions u + v and r u should make sense. Certain restrictions apply. For instance, u + v = v + u , 2 u + 3 u = 5 u . That is, addition and scalar multiplication in V should be like those of ndimensional vectors. Abstract vector space: definition Vector space is a set V equipped with two operations α : V × V → V and μ : R × V → V that have certain properties (listed below)....
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This note was uploaded on 11/08/2011 for the course MATH 304 taught by Professor Hobbs during the Spring '08 term at Texas A&M.
 Spring '08
 HOBBS
 Linear Algebra, Algebra

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