Rank in Matrices
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Rank
Maximum number of
Independent Columns
Number of Pivot
Column
Number of Non-zero
rows
Rank = ?
Rank = ?
3
3
Rank R = Rank A
Rank
Number of Pivot
Column
Number of Non-zero
rows
Maximum number of
Independent Columns
Rank
3 X 4
A matrix set has 4 vectors
belonging to R
3
is dependent
Matrix A is
full rank
if Rank A = min(m,n)
Matrix A is
rank deficient
if Rank A < min(m,n)
In R
m
, you cannot find more than m vectors that are independent.
Basic, Free Variables v.s. Rank
3 useful
equations
3 basic variables
2 free variables
non-zero row
rank
No. column –
non-zero row
nullity
=
=
=
=
Rank
Maximum number of
Independent Columns
Number of Pivot
Column
Number of Non-zero
rows of RREF
Number of Basic
Variables
Rank
Number of Free
Variables
Nullity = no. column - rank
Number of zero rows
of RREF
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.
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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
