{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lect17 - Chapter 3 Vector Spaces Math1111 Basis and...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 0 2 1 2 1 1 1 0 1 0 0 1 .
Background image of page 1

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

View Full Document Right Arrow Icon
Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 0 2 1 2 1 1 1 0 1 0 0 1 . Ans . In this case, N ( A ) = { x 5 : A x = 0 } The solutions of A x = 0 are x = α u + β v , where u = ( - 1 0 1 0 0 ) T and v = ( 0 - 1 0 1 1 ) T .
Background image of page 2
Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 0 2 1 2 1 1 1 0 1 0 0 1 . Ans . In this case, N ( A ) = { x 5 : A x = 0 } The solutions of A x = 0 are x = α u + β v , where u = ( - 1 0 1 0 0 ) T and v = ( 0 - 1 0 1 1 ) T . Apply elementary row operations: 1 2 1 0 2 1 2 1 1 1 0 1 0 0 1 -→ ······ -→ 1 0 1 0 0 0 1 0 0 1 0 0 0 1 - 1
Background image of page 3

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

View Full Document Right Arrow Icon
Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 0 2 1 2 1 1 1 0 1 0 0 1 . Ans . In this case, N ( A ) = { x 5 : A x = 0 } The solutions of A x = 0 are x = α u + β v , where u = ( - 1 0 1 0 0 ) T and v = ( 0 - 1 0 1 1 ) T . N ( A ) = Span ( u , v ) .
Background image of page 4
Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 0 2 1 2 1 1 1 0 1 0 0 1 . Ans . In this case, N ( A ) = { x 5 : A x = 0 } The solutions of A x = 0 are x = α u + β v , where u = ( - 1 0 1 0 0 ) T and v = ( 0 - 1 0 1 1 ) T . N ( A ) = Span ( u , v ) . Note: u , v are linearly independent. Exercise
Background image of page 5

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

View Full Document Right Arrow Icon
Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 0 2 1 2 1 1 1 0 1 0 0 1 . Ans . In this case, N ( A ) = { x 5 : A x = 0 } The solutions of A x = 0 are x = α u + β v , where u = ( - 1 0 1 0 0 ) T and v = ( 0 - 1 0 1 1 ) T . N ( A ) = Span ( u , v ) . Note: u , v are linearly independent. Exercise { u , v } is a basis for N ( A ) , so dim N ( A ) = 2 .
Background image of page 6
Chapter 3. Vector Spaces Math1111 Basis and Dimension Examples (Cont’d) Example . Find dim N ( A ) if A = 1 2 1 0 2 1 2 1 1 1 0 1 0 0 1 . Ans . In this case, N ( A ) = { x 5 : A x = 0 } The solutions of A x = 0 are x = α u + β v , where u = ( - 1 0 1 0 0 ) T and v = ( 0 - 1 0 1 1 ) T . N ( A ) = Span ( u , v ) . Note: u , v are linearly independent. Exercise { u , v } is a basis for N ( A ) , so dim N ( A ) = 2 . Observation dim N ( A ) = number of free variables in the solution of A x = 0
Background image of page 7

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

View Full Document Right Arrow Icon
Chapter 3. Vector Spaces Math1111 Vector Spaces Homework 9 Reading Textbook - p.148-150 Homework 9 Chapter 3 - Section 4 Exercises: Qn. 9, 10, 11, 12, 14, 16, 17, 18
Background image of page 8
Chapter 3. Vector Spaces Math1111 Change of Basis Motivation Question What is coordinate? For example, what is ( 4,3 ) in 2 meant?
Background image of page 9

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

View Full Document Right Arrow Icon
Chapter 3. Vector Spaces Math1111 Change of Basis Motivation Question What is coordinate? For example, what is ( 4,3 ) in 2 meant? As { e 1 , e 2 } is a basis of 2 , we may represent it as 4 e 1 + 3 e 2 .
Background image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}