lecture11 - Lecture 11 Fundamental Theorems of Linear...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 11 Fundamental Theorems of Linear Algebra Orthogonalily and Projection Shang-Hua Teng
Image of page 1

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

View Full Document Right Arrow Icon
The Whole Picture Rank(A) = m = n A x = b has unique solution [ ] I R = [ ] F I R = = 0 I R = 0 0 F I R Rank(A) = m < n A x = b has n-m dimensional solution Rank(A) = n < m A x = b has 0 or 1 solution Rank(A) < n, Rank(A) < m A x = b has 0 or n-rank(A) dimensions
Image of page 2
Basis and Dimension of a Vector Space A basis for a vector space is a sequence of vectors that The vectors are linearly independent The vectors span the space: every vector in the vector can be expressed as a linear combination of these vectors
Image of page 3

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

View Full Document Right Arrow Icon
Basis for 2D and n-D (1,0), (0,1) (1 1), (-1 –2) The vectors v 1 ,v 2 ,…v n are basis for R n if and only if they are columns of an n by n invertible matrix
Image of page 4
Column and Row Subspace C(A): the space spanned by columns of A Subspace in m dimensions The pivot columns of A are a basis for its column space Row space: the space spanned by rows of A Subspace in n dimensions The row space of A is the same as the column space of A T , C(A T ) The pivot rows of A are a basis for its row space The pivot rows of its Echolon matrix R are a basis for its row space
Image of page 5

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

View Full Document Right Arrow Icon
Important Property I: Uniqueness of Combination The vectors v 1 ,v 2 ,…v n are basis for a vector space V, then for every vector v in V, there is a unique way to write v as a combination of v 1 ,v 2 ,…v n .
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern