Understanding Rank in Matrices
Rank in matrices represents the maximum number of independent columns, with implications for pivot columns, basic variables, and free variables. The rank of a matrix is essential for determining its properties and dependencies. Learn about rank-deficient matrices, basic versus free variables, and more in this comprehensive guide.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
Presentation Transcript
More about Rank Rank
Rank Rank R = Rank A Maximum number of Independent Columns = 3 Rank = ? Number of Pivot Column = Number of Non-zero rows Rank = ? 3
Rank Maximum number of Independent Columns Rank A Number of columns = Rank A Min( Number of columns, Number of rows) Number of Pivot Column = Number of Non-zero rows Rank A Number of rows
Matrix A is full rank if Rank A = min(m,n) Rank Matrix A is rank deficient if Rank A < min(m,n) Given a mxn matrix A: Rank A min(m, n) Because the columns of A are independent is equivalent to rank A = n If m < n, the columns of A is dependent. 3 X 4 , , , A matrix set has 4 vectors belonging to R3 is dependent Rank A 3 In Rm, you cannot find more than m vectors that are independent.
Basic, Free Variables v.s. Rank ?? = ? 3 useful equations RREF(?) ? ? ? = rank non-zero row = 3 basic variables No. column non-zero row nullity 2 free variables = =
Rank Number of Pivot Column Maximum number of Independent Columns Rank Number of Basic Variables Number of Non-zero rows of RREF Nullity = no. column - rank Number of zero rows of RREF Number of Free Variables
