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Linear system, REF, RREF, nullspace

22 Pages

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Sophie Chrysostomou

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Linear Systems and Matrices
Linear Systems
DEFINITION: An m×nlinear system of equations is asystem of mlinear equations in n
variables: a11x1+a12x2+... +a1nxn=b1
a21x1+a22x2+... +a2nxn=b2
am1x1+am2x2+... +amnxn=bm
where x1,· · · xnarethe unknowns and aij are constantreal numbersforalli=1,2,· · · ,m,
and j=1,2,· · · n.
EXAMPLE: 2x1+3x2=9
In order to solvesuchasystem weperformin some sequence the following three operations:
1. Switchtwoequations.
2. Multiply an equation byanonzero constant.
3. Replace an equation bythatequation plus amultiple ofanother equation.
1. An m×nmatrix is anordered rectangular array,of realnumbers,with mrows and n
a11 a12 · · · a1n
a21 a22 · · · a2n
am1am2· · · amn
2. Wemaydenote the matrix bygiving it aname,sayA,and write
A=[aij ], where aij is the entry in the ith rowand jth column of A.
3. An m×1matrix
is calledacolumn vectorinRm.
4. A1×nmatrix c1c2· · · cnis calledarow vectorin Rn.
5. Amatrix is called asquarematrix if the number ofits rowsand the number ofits
columns are equal.
6. Asquare matrix Ais calledadiagonal matrix if
aij =0forall i6=j
7. Asquare matrix Ais calledan identity matrix if
aij =0for all i6=j
1for all i=j
8. Asquare matrix Ais calledan upper triangular matrix if
aij =0forall i>j
9. Asquare matrix Ais calledalower triangular matrix if
aij =0forall i<j
2011 bySophie Chrysostomou
Addition Subtractionand Scalar Multiplication
DEFINITION: Let A=[aij ]and B=[bij ]bem×nmatrices and kbeascalarthen:
i) A+Bis defined to bethe matrix withijth entry (A+B)ij =aij +bij
ii) ABis the matrix with ijth entry (AB)ij =aij bij
iii) kAis the matrix with ijth entry (kA)ij =kaij
Amatrix with zero entries only is called, azeromatrix and is denoted byO.
3 2
2 4
and B=
1 3
.Find A+Band 2A3B.
2011 bySophie Chrysostomou

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Linear Systems and Matrices Linear Systems DEFINITION: An m n linear system of equations is a system of m linear equations in n variables: a11 +1a x 12.2. + a x 1n n = b 1 a x + a x + ... + a x = b 21 1 22 2 2n n 2 . . . . . . am1 x1+ a m2x 2 ... + a mn xn = b m where x ,1x arn the unknowns and aij are constant real numbers for all i = 1,2, ,m, and j = 1,2,n. EXAMPLE: 2x1+ 3x 2 = 9 x1 2x 2 = 1 In order to solve such a system we perform in some sequence the following three operations: 1. Switch two equations. 2. Multiply an equation by a nonzero constant. 3. Replace an equation by that equation plus a multiple of another equation. www.notesolution.comDEFINITION: 1. An m n matrix is an ordered rectangular array, of real numbers, with m rows and n columns. a11 a12 a1n a21 a22 a2n . . . . . . . . a a a m1 m2 mn 2. We may denote the matrix by giving it a name, say A, and write A = [a ], where a is the entry in the ihrow and j thcolumn of A. ij ij b1 b2 3. An m 1 matrix . is called a column vector in R . . bm 4. A 1 n matrix c1 c2 cn is called a row vector in R . 5. A matrix is called a square matrix if the number of its rows and the number of its columns are equal. 6. A square matrix A is called a diagonal matrix if aij 0 for all i j 7. A square matrix A is called an identity matrix if a = 0 for all i j ij 1 for all i = j 8. A square matrix A is called an upper triangular matrix if a = 0 for all i > j ij 9. A square matrix A is called a lower triangular matrix if aij 0 for all i < j 2 2011 by Sophie Chrysostomou
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