RREF and Its Applications in Linear Algebra
Explore the Reduced Row Echelon Form (RREF) and its significance in linear algebra. Learn about concepts like linear combinations, independence, rank, and span through practical examples and the Column Correspondence Theorem. Discover the role of RREF in transforming coefficient matrices, augmented matrices, and solving systems of equations.
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Presentation Transcript
What can we know from RREF? Hung-yi Lee
Reference Textbook: Chapter 1.6, 1.7
Outline RREF v.s. Linear Combination RREF v.s. Independent RREF v.s. Rank RREF v.s. Span
RREF v.s. Linear Combination
Column Correspondence Theorem RREF ? = ?? ?? ? = ?? ?? If ?? is a linear combination of other columns of A a5= a1+a4 ?? is a linear combination of the corresponding columns of R with the same coefficients r5= r1+r4 ?? is a linear combination of the corresponding columns of A with the same coefficients If ?? is a linear combination of other columns of R r3= 3r1-2r2 a3= 3a1-2a2
Column Correspondence Theorem - Example a1 a2 a3a4a5 a6 r1 r2r3r4 r5 r6 a2= 2a1 r2= 2r1 a5= a1+a4 r5= r1+r4
Column Correspondence Theorem Intuitive Idea ?1+ ?2= ?3 6 8 9 9 0 2 15 8 11 ? = ?1+ ?2= ?3 ?1+ ?2= ?3 6 8 3 9 0 15 8 4 9 8 6 2 0 9 11 8 15 ? = ? = 7 ?1+ ?2= ?3 12 8 9 18 0 2 30 8 11 ? = Column Correspondence Theorem (Column ) row elementary operation column
Column Correspondence Theorem Reason Before we start: RREF Coefficient Matrix: ? ? RREF ? ? ? ? Augmented Matrix: ? ? ? ?
Column Correspondence Theorem Reason The RREF of matrix A is R ?? = ? and ?? = ? have the same solution set? The RREF of augmented matrix ? ?? = ? and ?? = ? have the same solution set ? is ? ? The RREF of matrix A is R ?? = 0 and ?? = 0 have the same solution set If ? = 0, then ? = 0.
Column Correspondence Theorem Reason The RREF of matrix A is R, ?? = 0 and ?? = 0 have the same solution set a1 a2 a3a4a5 a6 r1 r2r3r4 r5 r6 2 1 0 0 0 0 2 1 0 0 0 0 a2= 2a1 r2= 2r1 ?? = 0 ?? = 0 ? = ? = -2a1+a2=0 -2r1+r2=0
Column Correspondence Theorem Reason The RREF of matrix A is R, ?? = 0 and ?? = 0 have the same solution set a1 a2 a3a4a5 a6 r1 r2r3r4 r5 r6 1 0 0 1 0 0 r5= r1+r4 a5= a1+a4 ?? = 0 ?? = 0 ? = ? = 1 1 0 1 1 0 r1-r4+r5=0 a1-a4+a5=0
How about Rows? Are there row correspondence theorem? NO ? ? ?? ?? ? ? ?? ?? ? ? ?? ?? ? ? ?? ?? ???? ??,??,??,?? ???? ??,??,??,?? = Are they the same?
Span of Columns ? = ?? ?? ? = ?? ?? ???? ??, ,?? ???? ??, ,?? Are they the same? The elementary row operations change the span of columns.
NOTE Original Matrix v.s. RREF Columns: The relations between the columns are the same. The span of the columns are different. Rows: The relations between the rows are changed. The span of the rows are the same.
Column Correspondence Theorem pivot columns Leading entries linear linear independent independent The pivot columns are linear independent.
Column Correspondence Theorem pivot columns Leading entries a2= 2a1 a5= a1+a4 a6= 5a1 3a3+2a4 r2= 2r1 r5= r1+r4 r6= 5r1 3r3+2r4 The non-pivot columns are the linear combination of the previous pivot columns.
Independent 3X3 All columns are independent Columns are linear independent Every column is a pivot column RREF 1 0 0 0 1 0 0 0 1 Every column in RREF(A) is standard vector. Identity matrix
Independent 4X3 All columns are independent Columns are linear independent Every column is a pivot column RREF 1 0 0 0 0 1 0 0 0 0 1 0 ? ? Every column in RREF(A) is standard vector.
Independent 3X4 All columns are independent Columns are linear independent Every column is a pivot column Cannot be a pivot column RREF 1 0 0 0 1 0 0 0 1 Every column in RREF(A) is standard vector.
Independent The columns are dependent ( ) Dependent or Independent? More than 3 vectors in R3must be dependent. More than m vectors in Rmmust be dependent.
Rank Maximum number of Independent Columns = 3 Rank = ? Number of Pivot Column = Number of Non-zero rows Rank = ? 3
Properties of Rank from RREF 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
Properties of Rank from RREF Matrix A is full rank 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 Equations
Consistent or not Given Ax=b, if the reduced row echelon form of [ A b ] is Consistent b is in the span of the columns of A Given Ax=b, if the reduced row echelon form of [ A b ] is inconsistent b is NOT in the span of the columns of A 0 ?1+ 0 ?2+ 0 ?3= 1
Consistent or not Ax =b is inconsistent (no solution) The RREF of [A b] is 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ? 0 0 Only the last column is non-zero ? 0 Rank A rank [A b] Need to know b
Consistent or not Ax =b is consistent for every b RREF of [A b] cannot have a row whose only non-zero entry is at the last column RREF of A cannot have zero row Rank A = no. of rows
e.g. Consistent or not 3 independent columns Ax =b is consistent for every b Rank A = no. of rows Every b is in the span of the columns of A= ?1 ?? Every b belongs to S????1, S????1, ,?? = ?? ,?? m independent vectors can span ?? More than m vectors in Rmmust be dependent.
m independent vectors can span ?? More than m vectors in Rmmust be dependent. ? Consider R2 ? ? yes independent
Full Rank: Rank = n & Rank = m 0 0 1 0 The size of A is mxn 1 0 Rank A = n 0 0 A is square or Ax = b has at most one solution The columns of A are linearly independent. ? ? All columns are pivot columns. RREF of A:
Full Rank: Rank = n & Rank = m 0 0 0 0 1 1 The size of A is mxn 0 1 Rank A = m 0 0 0 1 A is square or Every row of R contains a pivot position (leading entry). Ax = b always have solution (at least one solution) for every b in Rm. The columns of A generate Rm.
