선형대수학 입문 주니어세미나 Introduction to Linear Algebra for engineers and computer scientists 송홍엽 연세대학교 전기전자공학부 [email protected]
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선형대수학입문
주니어세미나
Introduction to Linear Algebrafor engineers and computer scientists
송홍엽
연세대학교전기전자공학부
통신공학/신호처리/네트워크이론/제어공학 /채널코딩/암호그리고컴퓨터공학전분야에서필수 도구로사용함.
위의 전공 분야는거의 대부분이선형대수학의기본 이론을바탕으로새로운연구가 진행됨.
예를 들자면,
모든 비선형 방정식은 선형화를 거쳐서 “선형 연립방정식”의 형태로 모델링
모든 실계수 미분방정식은 “선형 연립방정식”으로 모델링
모든 (통신/압축/제어/랜덤)신호처리 과정은 “선형변환”으로 모델링
관련된 “행렬”의 성질을 파악하고 이를 “변형(대각화)”하여처리
2
왜 선형대수학을 공부하는가
강의실 규정을 준수한다.
지각 없는 전출
무단결석이후수업에참여하지못함. – nonpass 공식사유서가 있는 경우 미리 혹은 결석 이후 즉시 공식서류를 포함하여 결석사유를 설명하는 결석계를 이메일로 제출할 것.
지각하는경우, 사유서를수업이후당일에제출. (이메일로)
당일에 제출하지 않으면 nonpass
가급적 1분전에강의실에도착할것.
중간시험 100점 (50/100 이상이어야계속 수강 가능함)
기말시험 100점 (총합 120/200 점 이상이어야 PASS)
3
성적평가 Pass 조건
첫 시간까지 hysong2013으로 카톡메시지를보낸다.
핸드폰전원 OFF, 컴퓨터를지참하지않는다 수업시간중핸드폰전원이울리거나컴퓨터를켜고사용하면 NONPASS
오직 필기장과펜/연필을지참한다. 들으면서필기하기
교재/부교재/강의노트지참하지않는다. 수업시간전에미리읽고공부한다.
지각하지않는다. 전자출결에서지각은 5분이후확인.
이경우당일까지 YSCEC메모를사용하여 “지각사유서”를보내야함.
카톡방에공지하는강의자료를반드시 “미리 공부하고”해당 수업시간에 연습문제를풀어 제출한다.
제출은반드시수업시작시간에한다.
수업시간에는이문제에대한풀이와질문을받는다.
4
강의실 규정
총 13시간 수업
1. Two different views of Simultaneous Linear Equations
2. Gauss Elimination and LDU Decomposition
3. Determinant of a square matrix
4. Vector Space over a Field
5. Linear Transformation and Matrices
6. Four fundamental subspaces of an m x n matrix
중간시험
7. Projection and Least Square Solution
8. Orthogonal Matrices and QR Decomposition
9. Orthogonal Subspaces
10. Eigenvalues and Eigenvectors
11. Similar and Congruence relations
12. Positive Definite Matrices
13. Singular Value Decomposition
기말시험
5
본 강좌의 구성
Week Month Calendar (목-월-수) Topic Semester Calendar
01 3 2,3,4,5,6,7,8 1. Simultaneous Linear Equations(3.2.) 개강(3.6. - 3.8.) 수강신청 확인 및 변경
02 3 9,10,11,12,13,14,15 2. Gauss Elimination and LDU Decomposition
03 3 16,17,18,19,20,21,22 3. Determinant of a square matrix
04 3 23,24,25,26,27,28,29 4. Vector Space over a Field
05 3/4 30, 31,1,2,3,4,5 5. Linear Transformation and Matrices(4.3. - 4.5.) 수강철회
06 4 6,7,8,9,10,11,12 6. Four fundamental subspaces of an m x n matrix(4.7.) 학기 1/3선
07 4 13,14,15,16,17,18,19 중간시험 (학과 스케줄) (4.17. - 4.22.)중간시험
08 4 20,21,22,23,24,25,26 7. Projection and Least Square Solution (4.17. - 4.22.)중간시험
09 4/5 27,28,29,30,1,2,3 8. Orthogonal/Unitary Matrices and QR Decomposition (5.3.) 석가탄신일
10 5 4,5,6,7,8,9,10 9. Orthogonal Subspaces (5.5.) 어린이날
11 5 11,12,13,14,15,16,17 10. Eigenvalues and Eigenvectors (5.15.) 학기 2/3 선
12 5 18,19,20,21,22,23,24 11. Normal matrices and Positive Definite Matrices
13 5 25,26,27,28,29,30,31 12. Similarity and Congruence
14 6 1,2,3,4,5,6,7 13. Singular Value Decomposition (6.6.) 현충일
15 6 8,9,10,11,12,13,14 6.12 기말시험 (6.8. - 6.21.) 자율학습 및 기말시험
16 6 15,16,17,18,19,20,21 (6.8. - 6.21.) 자율학습 및 기말시험
Calendar (plan-A)Week starts from Thursday to the next Wednesday
연세대학교 송홍엽 강의노트
선형대수학 입문
Lecture #1
1-1. Matrices1-2. Simultaneous Linear Equations
1
1-1. MATRICES
3
a Matrix of size 𝑚 × 𝑛 over 𝑭
𝑸 – rational numbers
𝑅 – real numbers
𝐶 – complex numbers
𝑍 – integers (ring)
𝑍𝑛 - integers mod 𝑛 (ring)
𝒁𝒑 - integers mod 𝒑
𝑭𝟐 - binary numbers
etc...
components are members of 𝑭
1 2 3 ....... j ..... ............ n23...i...
m
𝑎𝑖,𝑗= A = (𝑎𝑖,𝑗)
column (열) vector
row (행) vector
𝑎𝑖,𝑗 = (𝑖, 𝑗)-component of A
(i,j)원소, (i,j)성분is a member of 𝑭
fieldor
ring
2𝑥 − 1 = 0 over 𝒁𝟓
⇔ 2𝑥 = 1⇔ 𝑥 = 1/2 = 3
4
Matrix addition
Let A and B be two matrices over the same F
A+B is possible if both A and B have the same size.
A+B has the same size as A and B.
We say
A+B=C = (cij)
where cij=aij+bij for all i and j
addition of matrices A and B
addition defined on F
5
Matrix multiplication
Let A and B be two matrices over the same F
The multiplication AB (or A·B) is possible if the number of columns of A is the same as the number of rows of B.
We say
AB=C = (cij)
where cij=ai1b1j+ai2b2j+...+ainbnj
or
𝑐𝑖𝑗 =
𝑙=1
𝑛
𝑎𝑖𝑙𝑏𝑙𝑗
when n = number of columns of A
= number of rows of B
multiplication of matrices A and B
addition defined on F
multiplication defined on F
1 2 3 ....... ... ..... ............ n23...i...
m
1 2 3 ....... j ..... ........ k23...
...
n
A B
=
1 2 3 ....... j ... ..... k23...i...
m
C
𝑚 × 𝒏 𝒏 × 𝑘 𝑚 × 𝑘
cij
cij = ai1b1j + ai2b2j + ... + ainbnj
= “dot product” of i-th row of A and j-th column of B
7
Multiplication by a constant
1 2 3 ....... j ..... ............ n23...i...
m
𝑎𝑖,𝑗A = (𝑎𝑖,𝑗) =
1 2 3 ....... j ..... ............ n23...i...
m
𝑐𝑎𝑖,𝑗cA = c(𝑎𝑖,𝑗) = (c𝑎𝑖,𝑗) =
8
Some Properties
Zero matrix: all the components are 0.
Identity matrix: all the components are 0, except for the diagonal components which are all 1.
It is a square matrix.
A+B=B+A for any two same size matrices
AB≠BA in general
(A+B)+C=A+B+C=A+(B+C) if A+B, B+C are possible.
if A,B,C all have the same size
A(BC)=ABC=(AB)C if AB, BC are possible.
A(B+C)=AB+AC if AB, AC, B+C are possible.
0 0 00 0 0
0 00 0
1 00 1
9
연습문제 1-1
Over 𝐹5, repeat the above calculations.
Note here that -1=4, -2=3, ½ =3, ¼ =4, etc.
1-2. SIMULTANEOUS LINEAR EQUATIONS
10
11
Simultaneous Linear Equations2 x 2 example
Consider
2x - y = 1
x + y = 5
3x = 6
or x=2
substitute back to ②: y=3
①②
+
yes. x=2 and y=3 is a solution
Row view
Column view
12
Two views of S.L.E.
Row view
Rows are lines
Solutions (if any) are the intersections of these lines(rows)
Column view
Columns are weighted and summed
Solutions (if any) are the appropriate weights of the columns
13
Row View
Rows are lines
Solutions (if any) are the intersections of these lines(rows)
2x - y = 1
x + y = 5
y = 2x-1
y = -x+5
5
-1
solution (x=2, y=3)
2
3
0
x
yy = 2x-1
y = -x+5
in general,
(1) no solution when they are parallel
(2) many solutions when they coincide
(3) unique solution in all other cases
THEOREM 1.
𝑥 − 𝑦 = 0𝑥 − 𝑦 = 1
𝑥 − 𝑦 = 12𝑥 − 2𝑦 = 2
15
linear combinations of vectors
Definition: 21와
−11의 linear combination은 𝑥
21
+ 𝑦−11형태의 vector를뜻함.
여기서 𝑥와 𝑦는실수.
Q: 10와
01의 linear combination으로어떤벡터
𝑎𝑏를만들수있는가?
𝑥10
+𝑦01
=𝑎𝑏의 solution이존재하는값 𝑎, 𝑏를모두찾아라
Q: 11와
22의 linear combination으로어떤벡터
𝑎𝑏를만들수있는가?
𝑥11
+𝑦22
=𝑎𝑏의 solution이존재하는값 𝑎, 𝑏를모두찾아라
𝑥10+𝑦
01
=𝑥𝑦 therefore, for ANY 𝑎 and 𝑏, the solution is 𝑥 = 𝑎, 𝑦 = 𝑏.
ANSWER: 어떠한 값 𝑎, 𝑏가 주어져도10와
01의 linear combination으로 벡터
𝑎𝑏를 만들 수 있
으며, 만드는 방법은 유일하다.
𝑥11+𝑦
22
=𝑥 + 2𝑦𝑥 + 2𝑦
therefore, for ANY 𝑎 = 𝑏, the solution is 𝑥 = 𝑎 − 2𝑦 and any 𝑦.
ANSWER: 𝑎 = 𝑏 를 만족하는 값에 대해서만 가능하며, 벡터𝑎𝑏
=𝑎𝑎
=𝑏𝑏를 만드는 방법은 무한
히 많이 존재한다. 즉, 임의의 실수 𝑦와 𝑥 = 𝑎 − 2𝑦로 결정되는 𝑥로 만들 수 있다.
16
𝟑10+𝟒
01
=𝟑𝟒 1
0
01
𝟒01
𝟑10
𝟑𝟒
𝟎𝟎
−𝟐11+𝟑
22
=𝟒𝟒
𝟎𝟎
−𝟐11
𝟑22
𝟒𝟒
11
22
17
21와
−11의 linear combination으로임의의벡터
𝑎𝑏를만들수있는가?
𝑥21
+𝑦−11
=𝒂𝒃의 solution이존재하는가?
𝟎𝟎
𝟐𝟏
𝑥21
is the arrow from the origin
to ANY point on this red line
𝑦−11
is the arrow from the origin
to ANY point on this blue line
−𝟏𝟏
Their sum could end at ANY point on the plane
18
𝟎𝟎
𝟐𝟏
−𝟏𝟏
𝟎𝟎
span
Definition:
A span of a vector 𝑣1 is the set of all its linear combinations, i.e.,
{𝑎𝑣1| 𝑎 ∈ 𝑅 }.
A span of S={𝑣1,𝑣2,...}. is the set of all their linear combinations, i.e.,
span(S) = { 𝑖 𝑎𝑖𝑣𝑖 | all 𝑎𝑖 ∈ 𝑅 }
S={ 21
, −11
} → span(S) is the whole (𝑥, 𝑦) plane
S={ 11
, 22
} → span(S) = span 11
is the line satisfying 𝑦 = 𝑥
Watch the video: What are "vectors"?
https://youtu.be/TgKwz5Ikpc8
19
20
Column View
Columns are weighted and summed
Solutions (if any) are the appropriate weights (scales, multiples) of the columns
2x - y = 1
x + y = 5
⇔ x 21
+ y −11
= 15
0 x
y
-1 2
121
−11
221
3−11
15
2 21
+ 3 −11
= 15
solution !!
ax + by = c
column vectors
ax + by = c
column vectors
• no solution if c is not a linear combination of 𝑎 and 𝑏
• at least one solution if c belongs to SPAN of 𝑎 and 𝑏
many solution ↔ 𝑎, 𝑏 and 𝑐 are on the same line
unique solution↔ otherwise
0x
y
-1 2
121
=b
11/2
=a
15
=c is NOT a linear combination
of a and b
0x
y
-1 2
121
=b
11/2
=a
42
=c
↔ 𝑎 and 𝑏 are on the same line, but 𝑐 is not on this line
THEOREM 2.
𝑥 + 2𝑦 = 11
2𝑥 + 𝑦 = 5
𝑥 + 2𝑦 = 41
2𝑥 + 𝑦 = 2
연습문제 1-2
다음 선형연립방정식의 해의 존재 유무를 row-view와 column-view 관점에서 설명하라
(1) over the reals
2𝑥 + 3𝑦 = 13𝑥 + 2𝑦 = 0
(2) over the integers mod 5
2𝑥 + 3𝑦 = 13𝑥 + 2𝑦 = 0
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