linear algebra. 1.introduction determinant, matrix, n- dimension vector, linear equations, quadratic...
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Linear Algebra
• 1.Introduction• Determinant, matrix, n-dimension vector, linear equations, quadratic form. • The determinant and matrix are the most basic theories.
Leibniz introduced the definition of determinant
in 17th century。
Vandermonde is the first one to logicize about the determinant theory.
Cayley is recognized asthe founder of the matrix theory .
Linear Algebra Preface
• The matrix theory developed quickly during 20th century.
As a branch of math, it is used widely in physics, biology and economics.
• Matrix is more important than determinant in math.
• 2.Characters of the Course• Strong abstract and utility.
• Study the discrete variable.
• 3.Pedagogical Organization• Be primary for classroom teaching.
• Pay attention to explaining.
• Motivate to study,respond and exercise.
• 4.Requests of Study• Work hard at basic definitions.
• Think frequently, and explore bravely.
• Cultivate the ability.
• Listen carefully, and finish homework by yourself.
• 5.References• Explain and Answer to Linear Algebra Examples,
Exercises,Test Papers
Northwest University of Industry Publishing House
• Studying Directory during University——Linear Algebra
Shandong University Publishing House
Domoreexer-cises！
The Conception of Matrix
• 1.Definition of Matrix Equations
mnmnmm
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
2211
22222121
11212111
The coefficients can be recorded compactly in a rectangular array.
mnmm
n
n
aaa
aaa
aaa
21
22221
11211
This is a Matrix
An mn Matrix is a rectangular array of numbers in m rows and n columns, which is simply called Matrix.
The is called Matrix’s row, and the erect is called Matrix’s column. is the entry in the ith row, jth column of the Matrix.
ija
Matrix whose entries are all real is called a real Matrix.
We talk about real Matrix only.
We use capital letters A, B, C, etc, to denote Matrix, for example
mnmm
n
n
aaa
aaa
aaa
A
21
22221
11211
simplynmijaA )(
)( 11211 naaa
1
21
11
ma
a
a
Row Matrix
ColumnMatrixsubscript
nnnn
n
n
nn
aaa
aaa
aaa
A
21
22221
11211When m=n, in other words, the number of rows is equal to the number of columns,the Matrix is called square matrix.
main diagonal
Some Special forms of Matrix
00
00
.1
nmO
nna
a
11
.2
k
k
.3
1
1
.4 nE
nn
n
n
a
aa
aaa
222
11211
.5
nnnn aaa
aa
a
21
2221
11
6.Echelon Forms A matrix is said to be an echelon matrix,if 1)the zero rows,if any,are below all nonzero rows and 2)the first nonzero entry in any row is to the right of thefirst nonzero entry of the previous row.(This statement puts
no restriction on the first row)
They are all called echelon forms.
73325
00321
00069
00001
0022
0001
0000
00000
08700
54321
10000
98000
12210
312075
00000
00432
00605
00001
00001
00321
12344
Are they echelon forms?No!Please remember the characters of echelon form,
and the definition of it.
Echelon form is common in application!
Operations of matrices
I. Linear Operations1.Equality: two matrices are equal if they have the same size(i.e. the same numbers of rows and columns, and the corresponding entries are equal.) That is
nmijaA
nmijbB
=
same sizeijij ba
Corresponding entries are equal.
2.Addition , Subtraction
nmijaA
nmijbB
Let matrices
and define
nmijij baBA )( nmijij baBA )(
Obviously, A+B=B+A (A+B)+C=A+(B+C)
A+0=0+A=A A-A=0
Negative Matrix
nmijaA
Whose negative matrix is
The negative of A is written as –A, and that is
nmijaA
nmija
3.Scalar Multiplication
mnmm
n
n
kakaka
kakaka
kakaka
21
22221
11211
is called scalar multiplication,and it`s written as kA.
kA
1k A 1k A AA 1 OoA
kBkABAk
lAkAAlkAkllAk
)(
,)(,)()(
Multiplication of Matrix
3132121111 xaxaxay 3232221212 xaxaxay
2321313
2221212
2121111
tbtbx
tbtbx
tbtbxand
2321322121211
13113211211111
)(
)(
tbababa
tbababay
2322322221221
13123212211212
)(
)(
tbababa
tbababay
232221
131211
aaa
aaaA
3231
2221
1211
bb
bb
bb
B
322322221221312321221121
321322121211311321121111
babababababa
babababababa
232221
131211
aaa
aaa
3231
2221
1211
bb
bb
bb
smijaA )( nsijbB )(
Generally,nmijc )(
sjisjijiij bababac 2211
=
ABC
)( 21 isii aaa
sj
j
j
b
b
b
2
1
ijc
nssmnm BAC
11
11,
11
111e.g. BA：
AB0
00
0= O
22
22BA
BAAB Obviously, This is the difference between
matrices and numbers
11
01,
12
41,
63
42：e.g.2 CBA
69
46,
69
46ACAB ACAB
CB BUT,
This is also the difference between
matrices and numbers
Remember：1.Do not apply commutative law；2.Do not apply cancellation law；3.Have a nonzero null divisor.
nnmnmm EAAAE
kBABkAABk
CABAACB
ACABCBA
BCACAB
.4
)()()(.3
)(
)(.2
)().(1
BAABBA
thensize,sametheis,Let.5
AB BA
Please pay attention to character 5, if the size of A isnot equal to B`, the result is wrong.
right? that ismnnmmnnm BABA False!
Positive Integer Powers of Matrix
AAAAk EA 0 lklk AAA
kkk BAAB )( QUESTIONkkk BAAB )(
Transpose of a Matrix
nmijaA
mnjia
TA Aor
TT
TTT
TT
kAkA
BABA
AA
)(
)(
)( TAB)( TT AB
This is the point!
smijaA
nsijbB
nmijcABC
mnijTT dAB )(
msji
T aA
snji
T bB
sijsijijji bababac 2211
jssijijiij abababd 2211
jic ijd=
That is TTT ABAB )(TTTT ABCABC )( ?
11
T
nn
T
a
a
jic ijd=
Symmetric and Antisymmetric matrices
AAT :Matrix Symmetric
AAT :MatrixricAntisymmet
TTT AAAAAA ,,TAA
22
TT AAAAA
Any square matrix can be the sum of a symmetric matrixand an antisymmetric matrix.
The determinant of odd number order antisymmetric matrix is equal to zero.
jiij aa 0 and iijiij aaa
?
068
602
820
0
e.g.1:let A and B are both m*n matrices，prove that AB is a symmetric matrix if and only if AB=BA.
Prove： : TAB)( ABTTT ABAB )( and BA
BAAB
: BAAB TTT ABAB )( BA AB
matrix. symmetric a isAB
e.g.2： To find the power of
cossin
sincosA
?nA
22
22
sincoscossin2
cossin2sincos
2cos2sin
2sin2cos
cossin
sincos
cossin
sincos2A
)1cos()1sin(
)1sin()1cos(let 1
nn
nnAn
cossin
sincos
)1cos()1sin(
)1sin()1cos(then 1
nn
nnAAA nn
nn
nn
cossin
sincos
nn
nnAn
cossin
sincos