1 of 29

## What's hot

02 linear algebra
02 linear algebraRonald Teo

System Of Linear Equations
System Of Linear Equationssaahil kshatriya

Machine Learning-Linear regression
Machine Learning-Linear regressionkishanthkumaar

Support vector machines (svm)
Support vector machines (svm)Sharayu Patil

2. Linear Algebra for Machine Learning: Basis and Dimension
2. Linear Algebra for Machine Learning: Basis and DimensionCeni Babaoglu, PhD

Eigen values and eigen vectors engineering
Eigen values and eigen vectors engineeringshubham211

Gauss Elimination & Gauss Jordan Methods in Numerical & Statistical Methods
Gauss Elimination & Gauss Jordan Methods in Numerical & Statistical MethodsJanki Shah

Eigen value and eigen vector
Eigen value and eigen vectorRutvij Patel

ML - Multiple Linear Regression
ML - Multiple Linear RegressionAndrew Ferlitsch

Mathematics Foundation Course for Machine Learning & AI By Eduonix
Mathematics Foundation Course for Machine Learning & AI By Eduonix Nick Trott

Classification Based Machine Learning Algorithms
Classification Based Machine Learning AlgorithmsMd. Main Uddin Rony

Optimization in Deep Learning
Optimization in Deep LearningYan Xu

Linear Algebra – A Powerful Tool for Data Science
Linear Algebra – A Powerful Tool for Data SciencePremier Publishers

Eigenvalues and Eigenvectors
Eigenvalues and EigenvectorsVinod Srivastava

Matrices and System of Linear Equations ppt
Matrices and System of Linear Equations pptDrazzer_Dhruv

### What's hot(20)

02 linear algebra
02 linear algebra

System Of Linear Equations
System Of Linear Equations

Machine Learning-Linear regression
Machine Learning-Linear regression

Support vector machines (svm)
Support vector machines (svm)

2. Linear Algebra for Machine Learning: Basis and Dimension
2. Linear Algebra for Machine Learning: Basis and Dimension

Eigen values and eigen vectors engineering
Eigen values and eigen vectors engineering

Gauss Elimination & Gauss Jordan Methods in Numerical & Statistical Methods
Gauss Elimination & Gauss Jordan Methods in Numerical & Statistical Methods

Eigen value and eigen vector
Eigen value and eigen vector

Numerical analysis ppt
Numerical analysis ppt

ML - Multiple Linear Regression
ML - Multiple Linear Regression

Mathematics Foundation Course for Machine Learning & AI By Eduonix
Mathematics Foundation Course for Machine Learning & AI By Eduonix

Classification Based Machine Learning Algorithms
Classification Based Machine Learning Algorithms

Matrix_PPT.pptx
Matrix_PPT.pptx

Optimization in Deep Learning
Optimization in Deep Learning

Echelon forms
Echelon forms

Linear Algebra – A Powerful Tool for Data Science
Linear Algebra – A Powerful Tool for Data Science

Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors

Jacobi method
Jacobi method

Matrices and System of Linear Equations ppt
Matrices and System of Linear Equations ppt

Matrices ppt
Matrices ppt

## Similar to 1. Linear Algebra for Machine Learning: Linear Systems

Matrix and It's Applications
Matrix and It's ApplicationsPritom Chaki

Beginning direct3d gameprogrammingmath05_matrices_20160515_jintaeks
Beginning direct3d gameprogrammingmath05_matrices_20160515_jintaeksJinTaek Seo

Matrix and Determinants
Matrix and DeterminantsAarjavPinara

Bba i-bm-u-2- matrix -
Bba i-bm-u-2- matrix -Rai University

Linear Algebra Presentation including basic of linear Algebra
Linear Algebra Presentation including basic of linear AlgebraMUHAMMADUSMAN93058

Linear Algebra and Matrix
Linear Algebra and Matrixitutor

Appendix B Matrices And Determinants
Appendix B Matrices And DeterminantsAngie Miller

5. Linear Algebra for Machine Learning: Singular Value Decomposition and Prin...
5. Linear Algebra for Machine Learning: Singular Value Decomposition and Prin...Ceni Babaoglu, PhD

Solution to linear equhgations
Solution to linear equhgationsRobin Singh

Matrices and determinants_01
Matrices and determinants_01nitishguptamaps

systems of linear equations & matrices
systems of linear equations & matricesStudent

matrix-algebra-for-engineers (1).pdf
matrix-algebra-for-engineers (1).pdfShafaqMehmood2

### Similar to 1. Linear Algebra for Machine Learning: Linear Systems(20)

Matrix and It's Applications
Matrix and It's Applications

Beginning direct3d gameprogrammingmath05_matrices_20160515_jintaeks
Beginning direct3d gameprogrammingmath05_matrices_20160515_jintaeks

Matrix and Determinants
Matrix and Determinants

Bba i-bm-u-2- matrix -
Bba i-bm-u-2- matrix -

Linear Algebra Presentation including basic of linear Algebra
Linear Algebra Presentation including basic of linear Algebra

Linear Algebra and Matrix
Linear Algebra and Matrix

Appendix B Matrices And Determinants
Appendix B Matrices And Determinants

Maths
Maths

Matrix
Matrix

5. Linear Algebra for Machine Learning: Singular Value Decomposition and Prin...
5. Linear Algebra for Machine Learning: Singular Value Decomposition and Prin...

Solution to linear equhgations
Solution to linear equhgations

Linear Algebra and its use in finance:
Linear Algebra and its use in finance:

Linear algebra
Linear algebra

Presentation on matrix
Presentation on matrix

Matrices and determinants_01
Matrices and determinants_01

systems of linear equations & matrices
systems of linear equations & matrices

Rankmatrix
Rankmatrix

Matrices & Determinants.pdf
Matrices & Determinants.pdf

1560 mathematics for economists
1560 mathematics for economists

matrix-algebra-for-engineers (1).pdf
matrix-algebra-for-engineers (1).pdf

ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...
ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...JojoEDelaCruz

ICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfVanessa Camilleri

ROLES IN A STAGE PRODUCTION in arts.pptx
ROLES IN A STAGE PRODUCTION in arts.pptxVanesaIglesias10

4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptxmary850239

Congestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentationdeepaannamalai16

GRADE 4 - SUMMATIVE TEST QUARTER 4 ALL SUBJECTS
GRADE 4 - SUMMATIVE TEST QUARTER 4 ALL SUBJECTSJoshuaGantuangco2

TEACHER REFLECTION FORM (NEW SET........).docx

HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...Nguyen Thanh Tu Collection

Concurrency Control in Database Management system
Concurrency Control in Database Management systemChristalin Nelson

Millenials and Fillennials (Ethical Challenge and Responses).pptx
Millenials and Fillennials (Ethical Challenge and Responses).pptxJanEmmanBrigoli

Karra SKD Conference Presentation Revised.pptx
Karra SKD Conference Presentation Revised.pptxAshokKarra1

ANG SEKTOR NG agrikultura.pptx QUARTER 4
ANG SEKTOR NG agrikultura.pptx QUARTER 4MiaBumagat1

Expanded definition: technical and operational
Expanded definition: technical and operationalssuser3e220a

INCLUSIVE EDUCATION PRACTICES FOR TEACHERS AND TRAINERS.pptx
INCLUSIVE EDUCATION PRACTICES FOR TEACHERS AND TRAINERS.pptx

ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...
ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...

ICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdf

ROLES IN A STAGE PRODUCTION in arts.pptx
ROLES IN A STAGE PRODUCTION in arts.pptx

4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptx

Congestive Cardiac Failure..presentation
Congestive Cardiac Failure..presentation

GRADE 4 - SUMMATIVE TEST QUARTER 4 ALL SUBJECTS
GRADE 4 - SUMMATIVE TEST QUARTER 4 ALL SUBJECTS

TEACHER REFLECTION FORM (NEW SET........).docx
TEACHER REFLECTION FORM (NEW SET........).docx

HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...

Concurrency Control in Database Management system
Concurrency Control in Database Management system

Millenials and Fillennials (Ethical Challenge and Responses).pptx
Millenials and Fillennials (Ethical Challenge and Responses).pptx

Karra SKD Conference Presentation Revised.pptx
Karra SKD Conference Presentation Revised.pptx

ANG SEKTOR NG agrikultura.pptx QUARTER 4
ANG SEKTOR NG agrikultura.pptx QUARTER 4

Paradigm shift in nursing research by RS MEHTA
Paradigm shift in nursing research by RS MEHTA

Expanded definition: technical and operational
Expanded definition: technical and operational

### 1. Linear Algebra for Machine Learning: Linear Systems

• 1. Seminar Series on Linear Algebra for Machine Learning Part 1: Linear Systems Dr. Ceni Babaoglu Ryerson University cenibabaoglu.com Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 2. Overview 1 Matrices and Matrix Operations 2 Special Types of Matrices 3 Inverse of a Matrix 4 Determinant of a Matrix 5 A statistical Application: Correlation Coeﬃcient 6 Matrix Transformations 7 Systems of Linear Equations 8 Linear Systems and Inverses 9 References Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 3. Matrices An m × n matrix A =      a11 a12 a13 . . . a1n a21 a22 a23 . . . a2n ... ... ... ... ... am1 am2 am3 . . . amn      = [aij ] The i th row of A is A = ai1 ai2 ai3 . . . ain , (1 ≤ i ≤ m) The j th column of A is A =      a1j a2j ... amj      , (1 ≤ j ≤ n) Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 4. Matrix Operations Matrix Addition A + B = [aij ] + [bij ] , C = [cij ] cij = aij + bij , i = 1, 2, · · · , m, j = 1, 2, · · · , n. Scalar Multiplication rA = r [aij ] , C = [cij ] cij = r aij , i = 1, 2, · · · , m, j = 1, 2, · · · , n. Transpose of a Matrix AT = aT ij , aT ij = aji Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 5. Special Types of Matrices Diagonal Matrix An n × n matrix A = [aij ] is called a diagonal matrix if aij = 0 for i = j      a 0 . . . 0 0 1 . . . 0 ... ... ... ... 0 0 . . . 1      Identity Matrix The scalar matrix In = [dij ], where dii = 1 and dij = 0 for i = j, is called the n × n identity matrix      1 0 . . . 0 0 1 . . . 0 ... ... ... ... 0 0 . . . 1      Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 6. Special Types of Matrices Upper Triangular Matrix An n × n matrix A = [aij ] is called upper triangular if aij = 0 for i > j   2 b c 0 3 0 0 0 1   Lower Triangular Matrix An n × n matrix A = [aij ] is called lower triangular if aij = 0 for i < j   2 0 0 0 3 0 a b 1   Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 7. Special Types of Matrices Symmetrix Matrix A matrix A with real entries is called symmetric if AT = A.   1 b c b 2 d c d 3   Skew Symmetric Matrix A matrix A with real entries is called skew symmetric if AT = −A.   0 b −c −b 0 −d c d 0   Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 8. Matrix Operations Inner Product a · b = a1b1 + a2b2 + · · · + anbn = n i=1 ai bi Matrix Multiplication of an m × p matrix and p × n matrix cij = ai1b1j + ai2b2j + · · · + aipbpj = p k=1 aikbkj , 1 ≤ i ≤ m, 1 ≤ j ≤ n. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 9. Algebraic Properties of Matrix Operations Let A, B and C be matrices of appropriate sizes; r and s be real numbers. A + B is a matrix of the same dimensions as A and B. A + B = B + A A + (B + C) = (A + B) + C For any matrix A, there is a unique matrix 0 such that A + 0 = A. For each A, there is a unique matrix −A, A such that A + (−A) = O. A(BC) = (AB)C (A + B)C = AC + BC C(A + B) = CA + CB r(sA) = (rs)A (r + s)A = rA + sA r(A + B) = rA + rB A(rB) = r(AB) = (rA)B Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 10. Inverse of a Matrix Nonsingular Matrices An n × n matrix is called nonsingular, or invertible if there exists an n × n matrix B such that AB = BA = In. Inverse Matrix Such a B is called an inverse of A. If such a B does not exist, A is called singular, or noninvertible. The inverse of a matrix, if it exists, is unique. AA−1 = A−1 A = In AA−1 = 1 2 3 4 −2 1 3/2 −1/2 = −2 1 3/2 −1/2 1 2 3 4 = 1 0 0 1 Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 11. Determinant of a Matrix Associated with every square matrix A is a number called the determinant, denoted by det(A). For 2 × 2 matrices, the determinant is deﬁned as A = a b c d , det(A) = ad − bc A = 2 1 −4 −2 , det(A) = (2)(−2) − (1)(−4) = 0 Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 12. Properties of Determinants 1 If I is the identity, then det(I) = 1. 2 If B is obtained from A by interchanging two rows, then det(B) = −det(A). 3 If B is obtained from A by adding a multiple of one row of A to another row, then det(B) = det(A). 4 If B is obtained from A by multiplying a row of A by the number m, then det(B) = m det(A). 5 Determinant of an upper (or lower) triangular matrix is equal to the product of its diagonal entries. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 13. Determinant of an n × n matrix Minor Suppose that in an n × n matrix A we delete the ith row and jth column to obtain an (n − 1) × (n − 1) matrix. The determinant of this sub-matrix is called the (i, j)th minor of A and is denoted by Mij . Cofactor The number (−1)i+j Mij is called the (i, j)th cofactor of A and is denoted by Cij . Determinant Let A be an n × n matrix. Then det(A) can be evaluated by expanding by cofactors along any row or any column: det(A) = ai1Ci1 + ai2Ci2 + · · · + ainCin, 1 ≤ i ≤ n. or det(A) = a1j C1j + a2j C2j + · · · + anj Cnj , 1 ≤ j ≤ n. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 14. Example Let’s ﬁnd the determinant of the following matrix. A =   2 −3 1 4 0 −2 3 −1 −3   . If we expand cofactors along the ﬁrst row: |A| = (2)C11 + (−3)C12 + (1)C13 = 2(−1)1+1 0 −2 −1 −3 − 3(−1)1+2 4 −2 3 −3 + 1(−1)1+3 4 0 3 −1 = 2(−2) + 3(−6) + (−4) = −26. If we expand along the third column, we obtain |A| = (1)C13 + (−2)C23 + (−3)C33 = 1(−1)1+3 4 0 3 −1 − 2(−1)2+3 2 −3 3 −1 − 3(−1)3+3 2 −3 4 0 = −26. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 15. Angle between to vectors The length of n-vector v =        v1 v2 ... vn−1 vn        is deﬁnes as v = v2 1 + v2 2 + · · · + v2 n−1 + v2 n . The angle between the two nonzero vectors is determined by cos(θ) = u · v u v . −1 u · v u v 1, 0 θ π Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 16. A statistical application: Correlation Coeﬃcient Sample means of two attributes ¯x = 1 n n i=1 x, ¯y = 1 n n i=1 y Centered form xc = [x1 − ¯x x2 − ¯x · · · xn − ¯x]T yc = [y1 − ¯y y2 − ¯y · · · yn − ¯y]T Correlation coeﬃcient Cor(xc, yc) = xc · yc xc yc r = n i=1(xi − ¯x)(yi − ¯y) n i=1(xi − ¯x)2 n i=1(yi − ¯y)2 Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 17. Linear Algebra vs Data Science 1 Length of a vector 2 Angle between the two vectors is small 3 Angle between the two vectors is near π 4 Angle between the two vectors is near π/2 1 Variability of a variable 2 The two variables are highly positively correlated 3 The two variables are highly negatively correlated 4 The two variables are uncorrelated Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 18. Matrix Transformations If A is an m × n matrix and u is an n-vector, then the matrix product Au is an m-vector. A funtion f mapping Rn into Rm is denoted by f : Rn → Rm. A matrix transformation is a function f : Rn into Rm deﬁned by f (u) = Au. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 19. Example Let f : R2 → R2 be the matrix transformation deﬁned by f (u) = 1 0 0 −1 u. f (u) = f x y = 1 0 0 −1 x y = x −y This transformation performs a reﬂection with respect to the x-axis in R2. To see a reﬂection of a point, say (2,-3) 1 0 0 −1 2 −3 = 2 3 Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 20. Systems of Linear Equations A linear equation in variables x1, x2, . . . , xn is an equation of the form a1x1 + a2x2 + . . . + anxn = b. A collection of such equations is called a linear system: a11x1 + a12x2 + · · · + a1nxn = b1 a21x1 + a22x2 + · · · + a2nxn = b2 ... ... ... ... am1x1 + am2x2 + · · · + amnxn = bm Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 21. Systems of Linear Equations For the system of equations a11x1 + a12x2 + · · · + a1nxn = b1 a21x1 + a22x2 + · · · + a2nxn = b2 ... ... ... ... am1x1 + am2x2 + · · · + amnxn = bm Ax = b The augmented matrix:     a11 a12 a13 . . . a1n b1 a21 a22 a23 . . . a2n b2 . . . . . . . . . . . . . . . . . . am1 am2 am3 . . . amn bm     If b1 = b2 = · · · = bm = 0, the system is called homogeneous. Ax = 0 Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 22. Linear Systems and Inverses If A is an n × n matrix, then the linear system Ax = b is a system of n equations in n unknowns. Suppose that A is nonsingular. Ax = b A−1 (Ax) = A−1 b (A−1 A)x = A−1 b Inx = A−1 b x = A−1 b x = A−1b is the unique solution of the linear system. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 23. Solving Linear Systems A matrix is in echelon form if 1 All zero rows, if there are any, appear at the bottom of the matrix. 2 The ﬁrst nonzero entry from the left of a nonzero row is a 1. This entry is called a leading one of its row. 3 For each nonzero row, the leading one appears to the right and below any leading ones in preceding rows. 4 If a column contains a leading one, then all other entries in that column are zero.    1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1         1 0 0 0 1 3 0 1 0 0 5 2 0 0 0 1 2 0 0 0 0 0 0 0         1 2 0 0 3 0 0 1 0 2 0 0 0 0 0 0 0 0 0 0     Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 24. Solving Linear Systems An elementary row operation on a matrix is one of the following: 1 interchange two rows, 2 add a multiple of one row to another, and 3 multiply one row by a non-zero constant. Two matrices are row equivalent if one can be converted into the other through a series of elementary row operations. Every matrix is row equivalent to a matrix in echelon form. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 25. Solving Linear Systems If an augmented matrix is in echelon form, then the ﬁrst nonzero entry of each row is a pivot. The variables corresponding to the pivots are called pivot variables, and the other variables are called free variables. A matrix is in reduced echelon form if all pivot entries are 1 and all entries above and below the pivots are 0. A system of linear equations with more unknowns than equations will either fail to have any solutions or will have an inﬁnite number of solutions. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 26. Example: Let’s solve the following system. x1 − 3 x2 + x3 = 1 2 x1 + x2 − x3 = 2 4 x1 + 4 x2 − 2 x3 = 1 5 x1 − 8 x2 + 2 x3 = 5     1 −3 1 2 1 −1 4 4 −2 5 −8 2 1 2 1 5     R2−2R1→R2 R3−4R1→R3 R4−5R1→R4 −−−−−−−→     1 −3 1 0 7 −3 0 16 −6 0 7 −3 1 0 −3 0     R2/7→R2 −−−−−→     1 −3 1 0 1 −3/7 0 16 −6 0 7 −3 1 0 −3 0     R1+3R2→R1 R3−16R2→R3 R4−7R2→R4 −−−−−−−−→     1 0 −2/7 0 1 −3/7 0 0 6/7 0 0 0 1 0 −3 0     7R3/6→R3 −−−−−−→     1 0 −2/7 0 1 −3/7 0 0 1 0 0 0 1 0 −7/2 0     R1+2R3/7→R1 R2+3R3/7→R2 −−−−−−−−→     1 0 0 0 1 0 0 0 1 0 0 0 0 −3/2 −7/2 0     ⇔ x1 = 0, x2 = −3/2, x3 = −7/2 Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 27. Example: Let’s solve the following homogenous system. 2 x1 + 4 x2 + 3 x3 + 3 x4 + 3 x5 = 0 x1 + 2 x2 + x3 + 2 x4 + x5 = 0 x1 + 2 x2 + 2 x3 + x4 + 2 x5 = 0 x3 − x4 − x5 = 0     2 4 3 3 3 1 2 1 2 1 1 2 2 1 2 0 0 1 −1 −1 0 0 0 0     R1↔R2 −−−−→     1 2 1 2 1 2 4 3 3 3 1 2 2 1 2 0 0 1 −1 −1 0 0 0 0     R2−2R1→R2 R3−R1→R3 −−−−−−−→     1 2 1 2 1 0 0 1 −1 1 0 0 1 −1 1 0 0 1 −1 −1 0 0 0 0     R3−R2→R3 R4−R2→R4 −−−−−−−→     1 2 1 2 1 0 0 1 −1 1 0 0 0 0 0 0 0 0 0 −2 0 0 0 0     R3↔R4 −−−−→     1 2 1 2 1 0 0 1 −1 1 0 0 0 0 −2 0 0 0 0 0 0 0 0 0     −R3/2→R3 −−−−−−−→     1 2 1 2 1 0 0 1 −1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0     x1 + 2x2 + x3 + 2x4 + x5 = 0, x3 − x4 + x5 = 0 x5 = 0, x2 = α, x4 = β, x3 = β, x1 = −2α − β − 2β. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 28. Example: Let’s use elementary row operations to ﬁnd A−1 if A =   4 3 2 5 6 3 3 5 2  .   4 3 2 5 6 3 3 5 2 1 0 0 0 1 0 0 0 1   R1−R3→R1 −−−−−−−→   1 −2 0 5 6 3 3 5 2 1 0 −1 0 1 0 0 0 1   R2−5R1→R2 R3−3R1→R3 −−−−−−−→   1 −2 0 0 16 3 0 11 2 1 0 −1 −5 1 5 −3 0 4   R2/16→R2 −−−−−−→   1 −2 0 0 1 3/16 0 11 2 1 0 −1 −5/16 1/16 5/16 −3 0 4   R1+2R2→R1R3−11R1→R3 −−−−−−−−−−−−−−−→   1 0 3/8 0 1 3/16 0 0 −1/16 3/8 1/8 −3/8 −5/16 1/16 5/16 7/16 −11/16 9/16   R1+6R3→R1 R2+3R3→R2 −−−−−−−→   1 0 0 0 1 0 0 0 −1/16 3 −4 3 1 −2 2 7/16 −11/16 9/16   −16R3→R3 −−−−−−→   1 0 0 0 1 0 0 0 1 3 −4 3 1 −2 2 −7 11 −9   A−1 =   3 −4 3 1 −2 2 −7 11 −9   Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
• 29. References Linear Algebra With Applications, 7th Edition by Steven J. Leon. Elementary Linear Algebra with Applications, 9th Edition by Bernard Kolman and David Hill. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Linear Systems
Current LanguageEnglish
Español
Portugues
Français
Deutsche