Lect2-01-web

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

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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) 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 mn-dimensional 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 n-dimensional 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)....
View Full Document

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.

Page1 / 25

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

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online