Linear algebra for economists:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2011
|
| Series: | Springer texts in business and economics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 279 S. graph. Darst. |
| ISBN: | 9783642205699 9783642270024 |
Staff View
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV037315547 | ||
| 003 | DE-604 | ||
| 005 | 20140519 | ||
| 007 | t | ||
| 008 | 110401s2011 gw d||| |||| 00||| eng d | ||
| 015 | |a 11,N13 |2 dnb | ||
| 016 | 7 | |a 1010702920 |2 DE-101 | |
| 020 | |a 9783642205699 |c Gb. : ca. EUR 85.55 (DE) (freier Pr.), ca. sfr 115.00 (freier Pr.) |9 978-3-642-20569-9 | ||
| 020 | |a 9783642270024 |9 978-3-642-27002-4 | ||
| 035 | |a (OCoLC)725042706 | ||
| 035 | |a (DE-599)DNB1010702920 | ||
| 040 | |a DE-604 |b ger |e rakddb | ||
| 041 | 0 | |a eng | |
| 044 | |a gw |c XA-DE-BE | ||
| 049 | |a DE-M382 |a DE-355 |a DE-384 |a DE-N2 |a DE-M49 |a DE-19 | ||
| 082 | 0 | |a 512.502433 |2 22//ger | |
| 084 | |a QH 110 |0 (DE-625)141531: |2 rvk | ||
| 084 | |a QH 140 |0 (DE-625)141533: |2 rvk | ||
| 084 | |a SK 220 |0 (DE-625)143224: |2 rvk | ||
| 084 | |a 330 |2 sdnb | ||
| 084 | |a MAT 024f |2 stub | ||
| 084 | |a MAT 150f |2 stub | ||
| 084 | |a WIR 017f |2 stub | ||
| 100 | 1 | |a Aleskerov, Fuad T. |e Verfasser |0 (DE-588)170311910 |4 aut | |
| 245 | 1 | 0 | |a Linear algebra for economists |c Fuad Aleskerov ; Hasan Ersel ; Dmitri Piontkovski |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2011 | |
| 300 | |a XII, 279 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer texts in business and economics | |
| 650 | 0 | 7 | |a Wirtschaftsmathematik |0 (DE-588)4066472-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lineare Optimierung |0 (DE-588)4035816-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lineare Algebra |0 (DE-588)4035811-2 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
| 689 | 0 | 0 | |a Lineare Algebra |0 (DE-588)4035811-2 |D s |
| 689 | 0 | 1 | |a Lineare Optimierung |0 (DE-588)4035816-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Wirtschaftsmathematik |0 (DE-588)4066472-7 |D s |
| 689 | 1 | 1 | |a Lineare Algebra |0 (DE-588)4035811-2 |D s |
| 689 | 1 | |C b |5 DE-604 | |
| 700 | 1 | |a Ersel, Hasan |e Verfasser |0 (DE-588)171267664 |4 aut | |
| 700 | 1 | |a Piontkovski, Dmitri |e Verfasser |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-642-20570-5 |
| 856 | 4 | 2 | |m DNB Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022469871&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-022469871 | ||
Record in the Search Index
| _version_ | 1804145592470863872 |
|---|---|
| adam_text | IMAGE 1
CONTENTS
1 SOME BASIC CONCEPTS 1
1.1 INTRODUCTION 1
1.1.1 LINEARITY 1
1.1.2 SYSTEM OF COORDINATES ON THE PLANE R 2 2
1.2 MICROECONOMICS: MARKET EQUILIBRIUM 7
1.2.1 EQUILIBRIUM IN A SINGLE MARKET 8
1.2.2 MULTI-MARKET EQUILIBRIUM 9
1.3 MACROECONOMIC POLICY PROBLEM 10
1.3.1 A SIMPLE MACROECONOMIC POLICY MODEL WITH ONE TARGET 11
1.3.2 A MACROECONOMIC POLICY MODEL WITH MULTIPLE TARGETS AND MULTIPLE
INSTRUMENTS 13 1.4 PROBLEMS 15
2 VECTORS AND MATRICES 17
2.1 VECTORS 17
2.1.1 ALGEBRAIC PROPERTIES OF VECTORS 18
2.1.2 GEOMETRIC INTERPRETATION OF VECTORS AND OPERATIONS ON THEM 19
2.1.3 GEOMETRIC INTERPRETATION IN R 2 22
2.2 DOT PRODUCT OF TWO VECTORS 23
2.2.1 THE LENGTH OF A VECTOR, AND THE ANGLE BETWEEN TWO VECTORS 24
2.3 AN ECONOMIC EXAMPLE: TWO PLANTS 27
2.4 ANOTHER ECONOMIC APPLICATION: INDEX NUMBERS 29
2.5 MATRICES 30
2.5.1 OPERATIONS ON MATRICES 31
2.5.2 MATRIX MULTIPLICATION 32
2.5.3 TRACE OF A MATRIX 35
2.6 TRANSPOSE OF A MATRIX 35
2.7 RANKOFAMATRIX 36
2.8 ELEMENTARY OPERATIONS AND ELEMENTARY MATRICES 38
2.9 PROBLEMS 44
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1010702920
DIGITALISIERT DURCH
IMAGE 2
X CONTENTS
3 SQUARE MATRICES AND DETERMINANTS 49
3.1 TRANSFORMATION OF COORDINATES 49
3.1.1 TRANSLATION 49
3.1.2 ROTATION 50
3.2 SQUARE MATRICES 51
3.2.1 IDENTITY MATRIX 51
3.2.2 POWEROF A MATRIX AND POLYNOMIAL OF A MATRIX 52 3.3 SYSTEMS OF
LINEAR EQUATIONS: THE CASE OF TWO VARIABLES 52 3.4 DETERMINANT OF A
MATRIX 53
3.4.1 THE BASIC PROPERTIES OF DETERMINANTS 57
3.4.2 DETERMINANT AND ELEMENTARY OPERATIONS 60
3.5 PROBLEMS 61
4 INVERSE MATRIX 65
4.1 INVERSE MATRIX AND MATRIX DIVISION 65
4.2 RANK AND DETERMINANTS 70
4.3 PROBLEMS 71
5 SYSTEMS OF LINEAR EQUATIONS 75
5. 1 THE CASE OF UNIQUE SOLUTION: CRAMER S RULE 78
5.2 GAUSS METHOD: SEQUENTIAL ELIMINATION OF UNKNOWN VARIABLES 80 5.3
HOMOGENEOUS EQUATIONS 85
5.4 PROBLEMS 87
5.4.1 MATHEMATICAL PROBLEMS 87
5.4.2 ECONOMIC PROBLEMS 89
6 LINEAR SPACES 91
6. 1 LINEAR INDEPENDENCE OF VECTORS 92
6.1.1 ADDITION OF VECTORS AND MULTIPLICATION OF A VECTOR BY A REAL
NUMBER 96
6.2 ISOMORPHISM OF LINEAR SPACES 97
6.3 SUBSPACES 98
6.3.1 EXAMPLES OF SUBSPACES 98
6.3.2 A METHOD OF CONSTRUCTING SUBSPACES 99
6.3.3 ONE-DIMENSIONAL SUBSPACES 99
6.3.4 HYPERPLANE 99
6.4 COORDINATE CHANGE 100
6.5 ECONOMIC EXAMPLE: PRODUCTION TECHNOLOGY SET 101
6.6 PROBLEMS 104
7 EUCLIDEAN SPACES 107
7. 1 GENERAL DEFINITIONS 107
7.2 ORTHOGONAL BASES 109
7.3 LEAST SQUARES METHOD 117
7.4 ISOMORPHISM OF EUCLIDEAN SPACES 119
7.5 PROBLEMS 120
IMAGE 3
CONTENTS X
8 LINEAR TRANSFORMATIONS 123
8.1 ADDITION AND MULTIPLICATION OF LINEAR OPERATORS 130
8.2 INVERSE TRANSFORMATION, IMAGE AND KERNEL UNDER A TRANSFORMATION 132
8.3 LINEAR TRANSFORMATION MATRICES WITH RESPECT TO DIFFERENT BASES 135
8.4 PROBLEMS 137
9 EIGENVECTORS AND EIGENVALUES 141
9.1 MACROECONOMIC EXAMPLE: GROWTH AND CONSUMPTION 148 9.1.1 THEMODEL 148
9.1.2 NUMERICAL EXAMPLE 149
9.2 SELF-ADJOINT OPERATORS 150
9.3 ORTHOGONAL OPERATORS 153
9.4 QUADRATIC FORMS 156
9.5 PROBLEMS 161
10 LINEAR MODEL OF PRODUCTION IN A CLASSICAL SETTING 1 65
10.1 INTRODUCTION 165
10.2 THE LEONTIEF MODEL 169
10.3 EXISTENCE OF A UNIQUE NON-NEGATIVE SOLUTION TO THE LEONTIEF SYSTEM
172
10.4 CONDITIONS FOR GETTING A POSITIVE (ECONOMICALLY MEANINGFUL)
SOLUTION TO THE LEONTIEF MODEL 176
10.5 PRICES OF PRODUCTION IN THE LINEAR PRODUCTION MODEL 179 10.6
PERRON-FROBENIUS THEOREM 184
10.7 LINEAR PRODUCTION MODEL (CONTINUED) 187
10.7.1 SRAFFA SYSTEM: THE CASE OF BASIC COMMODITIES 189 10.7.2 SRAFFA
SYSTEM: NON-BASIC COMMODITIES ADDED 191 10.8 PROBLEMS 191
11 LINEAR PROGRAMMING 195
11.1 DIET PROBLEM 195
11.2 LINEAR PRODUCTION MODEL 197
11.3 CONVEXITY 200
11.4 TRANSPORTATION PROBLEM 206
11.5 DUAL PROBLEM 207
11.6 ECONOMIC INTERPRETATION OF DUAL VARIABLES 209
11.7 A GENERALIZATION OF THE LEONTIEF MODEL: MULTIPLE PRODUCTION
TECHNIQUES AND LINEAR PROGRAMMING 211 11.8 PROBLEMS 212
A NATURAL NUMBERS AND INDUCTION 217
A.I NATURAL NUMBERS: AXIOMATIC DEFINITION 217
A.2 INDUCTION PRINCIPLE 219
A.3 PROBLEMS 223
IMAGE 4
XUE CONTENTS
B METHODS OF EVALUATING DETERMINANTS 225
B. 1 TRANSFORMATION OF DETERMINANTS 225
B.2 METHODS OF EVALUATING DETERMINANTS OF HIGH ORDER 226 B.2.1 REDUCING
TO TRIANGULAR FORM 226
B.2.2 METHOD OF MULTIPLIERS 227
B.2.3 RECURSIVE DEFINITION OF DETERMINANT 228
B.2.4 REPRESENTATION OF A DETERMINANT AS A SUM OF TWO DETERMINANTS 230
B.2.5 CHANGING THE ELEMENTS OF DETERMINANT 230
B.2.6 TWOCLASSICAL DETERMINANTS 232
B.3 PROBLEMS 233
C COMPLEX NUMBERS 237
C. 1 OPERATIONS WITH COMPLEX NUMBERS 238
C. C. C. C.
C.
. 1 CONJUGATION 238
.2 MODULUS 239
.3 INVERSE AND DIVISION 239
.4 ARGUMENT 239
1.5 EXPONENT 241
C.2 ALGEBRAIC EQUATIONS 241
C.3 LINEAR SPACES OVER COMPLEX NUMBERS 244
C.4 PROBLEMS 245
D PSEUDOINVERSE 249
D. 1 DEFINITION AND BASIC PROPERTIES 250
D.I.I THE BASIC PROPERTIES OF PSEUDOINVERSE 251
D.2 FULL RANK FACTORIZATION AND A FORMULA FOR PSEUDOINVERSE 252 D.3
PSEUDOINVERSE AND APPROXIMATIONS 255
D.4 PROBLEMS 259
E ANSWERS AND SOLUTIONS 263
REFERENCES 275
INDEX 277
|
| any_adam_object | 1 |
| author | Aleskerov, Fuad T. Ersel, Hasan Piontkovski, Dmitri |
| author_GND | (DE-588)170311910 (DE-588)171267664 |
| author_facet | Aleskerov, Fuad T. Ersel, Hasan Piontkovski, Dmitri |
| author_role | aut aut aut |
| author_sort | Aleskerov, Fuad T. |
| author_variant | f t a ft fta h e he d p dp |
| building | Verbundindex |
| bvnumber | BV037315547 |
| classification_rvk | QH 110 QH 140 SK 220 |
| classification_tum | MAT 024f MAT 150f WIR 017f |
| ctrlnum | (OCoLC)725042706 (DE-599)DNB1010702920 |
| dewey-full | 512.502433 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.502433 |
| dewey-search | 512.502433 |
| dewey-sort | 3512.502433 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02330nam a2200577 c 4500</leader><controlfield tag="001">BV037315547</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20140519 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">110401s2011 gw d||| |||| 00||| eng d</controlfield><datafield tag="015" ind1=" " ind2=" "><subfield code="a">11,N13</subfield><subfield code="2">dnb</subfield></datafield><datafield tag="016" ind1="7" ind2=" "><subfield code="a">1010702920</subfield><subfield code="2">DE-101</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783642205699</subfield><subfield code="c">Gb. : ca. EUR 85.55 (DE) (freier Pr.), ca. sfr 115.00 (freier Pr.)</subfield><subfield code="9">978-3-642-20569-9</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783642270024</subfield><subfield code="9">978-3-642-27002-4</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)725042706</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)DNB1010702920</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakddb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">gw</subfield><subfield code="c">XA-DE-BE</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-M382</subfield><subfield code="a">DE-355</subfield><subfield code="a">DE-384</subfield><subfield code="a">DE-N2</subfield><subfield code="a">DE-M49</subfield><subfield code="a">DE-19</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">512.502433</subfield><subfield code="2">22//ger</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">QH 110</subfield><subfield code="0">(DE-625)141531:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">QH 140</subfield><subfield code="0">(DE-625)141533:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 220</subfield><subfield code="0">(DE-625)143224:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">330</subfield><subfield code="2">sdnb</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 024f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 150f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">WIR 017f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Aleskerov, Fuad T.</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)170311910</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Linear algebra for economists</subfield><subfield code="c">Fuad Aleskerov ; Hasan Ersel ; Dmitri Piontkovski</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Berlin [u.a.]</subfield><subfield code="b">Springer</subfield><subfield code="c">2011</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XII, 279 S.</subfield><subfield code="b">graph. Darst.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">Springer texts in business and economics</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Wirtschaftsmathematik</subfield><subfield code="0">(DE-588)4066472-7</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Lineare Optimierung</subfield><subfield code="0">(DE-588)4035816-1</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Lineare Algebra</subfield><subfield code="0">(DE-588)4035811-2</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="655" ind1=" " ind2="7"><subfield code="0">(DE-588)4123623-3</subfield><subfield code="a">Lehrbuch</subfield><subfield code="2">gnd-content</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Lineare Algebra</subfield><subfield code="0">(DE-588)4035811-2</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Lineare Optimierung</subfield><subfield code="0">(DE-588)4035816-1</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="689" ind1="1" ind2="0"><subfield code="a">Wirtschaftsmathematik</subfield><subfield code="0">(DE-588)4066472-7</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2="1"><subfield code="a">Lineare Algebra</subfield><subfield code="0">(DE-588)4035811-2</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2=" "><subfield code="C">b</subfield><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Ersel, Hasan</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)171267664</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Piontkovski, Dmitri</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Online-Ausgabe</subfield><subfield code="z">978-3-642-20570-5</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">DNB Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022469871&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-022469871</subfield></datafield></record></collection> |
| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV037315547 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T23:21:54Z |
| institution | BVB |
| isbn | 9783642205699 9783642270024 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022469871 |
| oclc_num | 725042706 |
| open_access_boolean | |
| owner | DE-M382 DE-355 DE-BY-UBR DE-384 DE-N2 DE-M49 DE-BY-TUM DE-19 DE-BY-UBM |
| owner_facet | DE-M382 DE-355 DE-BY-UBR DE-384 DE-N2 DE-M49 DE-BY-TUM DE-19 DE-BY-UBM |
| physical | XII, 279 S. graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer texts in business and economics |
| spelling | Aleskerov, Fuad T. Verfasser (DE-588)170311910 aut Linear algebra for economists Fuad Aleskerov ; Hasan Ersel ; Dmitri Piontkovski Berlin [u.a.] Springer 2011 XII, 279 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer texts in business and economics Wirtschaftsmathematik (DE-588)4066472-7 gnd rswk-swf Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Lineare Algebra (DE-588)4035811-2 s Lineare Optimierung (DE-588)4035816-1 s DE-604 Wirtschaftsmathematik (DE-588)4066472-7 s b DE-604 Ersel, Hasan Verfasser (DE-588)171267664 aut Piontkovski, Dmitri Verfasser aut Erscheint auch als Online-Ausgabe 978-3-642-20570-5 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022469871&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Aleskerov, Fuad T. Ersel, Hasan Piontkovski, Dmitri Linear algebra for economists Wirtschaftsmathematik (DE-588)4066472-7 gnd Lineare Optimierung (DE-588)4035816-1 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4066472-7 (DE-588)4035816-1 (DE-588)4035811-2 (DE-588)4123623-3 |
| title | Linear algebra for economists |
| title_auth | Linear algebra for economists |
| title_exact_search | Linear algebra for economists |
| title_full | Linear algebra for economists Fuad Aleskerov ; Hasan Ersel ; Dmitri Piontkovski |
| title_fullStr | Linear algebra for economists Fuad Aleskerov ; Hasan Ersel ; Dmitri Piontkovski |
| title_full_unstemmed | Linear algebra for economists Fuad Aleskerov ; Hasan Ersel ; Dmitri Piontkovski |
| title_short | Linear algebra for economists |
| title_sort | linear algebra for economists |
| topic | Wirtschaftsmathematik (DE-588)4066472-7 gnd Lineare Optimierung (DE-588)4035816-1 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Wirtschaftsmathematik Lineare Optimierung Lineare Algebra Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022469871&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT aleskerovfuadt linearalgebraforeconomists AT erselhasan linearalgebraforeconomists AT piontkovskidmitri linearalgebraforeconomists |