Column Correspondence Theorem and RREF Explained
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Column Correspondence
Theorem
RREF
a
5
=
a
1
+
a
4
r
5
=
r
1
+
r
4
r
3
=
3
r
1
-2
r
2
a
3
=
3
a
1
-2
a
2
Column Correspondence
Theorem - Example
a
1
a
2
a
3
a
4
a
5
a
6
r
1
r
2
r
3
r
4
r
5
r
6
r
2
= 2
r
1
a
2
= 2
a
1
Column Correspondence
Theorem - Example
r
5
=
r
1
+
r
4
a
5
=
a
1
+
a
4
a
1
a
2
a
3
a
4
a
5
a
6
r
1
r
2
r
3
r
4
r
5
r
6
r
2
= 2
r
1
a
2
= 2
a
1
Column Correspondence
Theorem – Reason
1
Elementary row operations
無法改變
column
間的關係
Switch rows
1 and 3
row 1 x 2
row 3 – row 1
Column Correspondence
Theorem – Reason
2
•
Basic concept
RREF
還是
RREF
…
不好說
Column Correspondence
Theorem – Reason
2
•
Basic concept
RREF
RREF
Augmented Matrix:
Coefficient Matrix:
A
R
Column Correspondence
Theorem – Reason 2
Column Correspondence
Theorem – Reason 2
r
2
= 2
r
1
a
1
a
2
a
3
a
4
a
5
a
6
r
1
r
2
r
3
r
4
r
5
r
6
-2
r
1
+
r
2
=
0
a
2
= 2
a
1
-2
a
1
+
a
2
=
0
Column Correspondence
Theorem – Reason 2
a
1
a
2
a
3
a
4
a
5
a
6
r
1
r
2
r
3
r
4
r
5
r
6
r
5
=
r
1
+
r
4
r
1
-
r
4
+
r
5
=0
a
5
=
a
1
+
a
4
a
1
-
a
4
+
a
5
=0
How about Rows?
•
Are there row correspondence theorem?
NO
This content delves into the Column Correspondence Theorem, illustrating how it relates to Reduced Row Echelon Form (RREF) and providing examples to enhance understanding. Discover the significance of elementary row operations, basic concepts of RREF, and the implications of matrices in solving systems of equations.
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Presentation Transcript
Column Column Correspondence Correspondence Theorem Theorem
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
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 Reason 1 Elementary row operations column ?1+ ?2= ?3 Switch rows 1 and 3 6 8 9 9 0 2 15 8 11 row 3 row 1 ? = ?1+ ?2= ?3 ?1+ ?2= ?3 6 8 3 9 0 15 8 4 9 8 6 2 0 9 11 8 15 ? = row 1 x 2 ? = 7 ?1+ ?2= ?3 12 8 9 18 0 2 30 8 11 ? =
Column Correspondence Theorem Reason 2 Basic concept RREF RREF
Column Correspondence Theorem Reason 2 Basic concept RREF ? ? ? ? Augmented Matrix: RREF Coefficient Matrix: ? ? A R ? ? ? ?
Column Correspondence Theorem Reason 2 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 2 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 2 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 ? ? ?? ?? ? ? ?? ?? ? ? ?? ?? ? ? ?? ??
