Understanding Matrix Rank and Variables
Explore the concept of matrix rank, independent columns, basic and free variables, and the relationship between them. Learn how to determine the rank, pivot columns, and nullity of a matrix using key equations and properties. Understand the significance of rank and variables in linear algebra and matrix manipulation.
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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
