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Lect2-01-web - MATH 304 Fall 2011 Linear Algebra Homework...

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MATH 304, Fall 2011 Linear Algebra
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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
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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
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MATH 304 Linear Algebra Lecture 8: Vector spaces. Subspaces.
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Linear operations on vectors Let x = ( x 1 , x 2 , . . . , x n ) and y = ( y 1 , y 2 , . . . , y n ) be n -dimensional 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 , 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 )
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Properties of linear operations x + y = y + x ( x + y ) + z = x + ( y + z ) x + 0 = 0 + x = x x + ( x ) = ( x ) + x = 0 r ( x + y ) = r x + r y ( r + s ) x = r x + s x ( rs ) x = r ( s x ) 1 x = x 0 x = 0 ( 1) x = x
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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 mn -dimensional vectors.
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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 n -dimensional vectors.
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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). The operation α is called addition . For any u , v V , the element α ( u , v ) is denoted u + v .
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