Mühendisler İçin
Lineer Cebir

Teori ve Python Uygulamaları

Prof. Dr. Zühal Küçükarslan Yüzbaşı ve Prof. Dr. Bahadır Yüzbaşı

Kitaptaki uygulamaları bölüm sırasıyla bulun. Örnek girdileri inceleyin ve hesapları kendi bilgisayarınızda çalıştırın.

Komutları ZIP arşivini açtığınız klasörde çalıştırın. Bazı Python dosyaları eski bölüm numaralarını taşır. Bu sayfadaki başlıklar güncel kitapla eşleşir.

Mühendislik Problemlerinin Lineer Dili

Mühendislikte Lineer Cebir ve Model Kurma

Üç sensörlü ölçüm modeli ve sabit öteleme.

Python dosyaları

Örnek verileri görüntüle
{
  "kitap_bolumu": 1,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "uc_sensor": {
      "kaynak": "bolum01_model_kurma.sensor_case",
      "state_names": [
        "temperature_change",
        "pressure_change",
        "humidity_change"
      ],
      "state_units": [
        "degC_difference",
        "kPa",
        "percent_RH_difference"
      ],
      "output_names": [
        "sensor_1",
        "sensor_2",
        "sensor_3"
      ],
      "output_units": [
        "V",
        "V",
        "V"
      ],
      "coefficient_units": [
        [
          "V/degC_difference",
          "V/kPa",
          "V/percent_RH_difference"
        ],
        [
          "V/degC_difference",
          "V/kPa",
          "V/percent_RH_difference"
        ],
        [
          "V/degC_difference",
          "V/kPa",
          "V/percent_RH_difference"
        ]
      ],
      "sensitivity_matrix": [
        [
          0.08,
          0.01,
          -0.002
        ],
        [
          0.015,
          0.12,
          0.004
        ],
        [
          -0.01,
          0.02,
          0.05
        ]
      ],
      "state_change": [
        10.0,
        5.0,
        20.0
      ],
      "offset_v": [
        0.2,
        -0.1,
        0.05
      ],
      "state_scale": [
        10.0,
        5.0,
        20.0
      ],
      "valid_absolute_change": [
        12.0,
        8.0,
        25.0
      ]
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum01_model_kurma.py

Vektörler, Ölçüm ve Geometri

Rüzgâr altında drone hareketi ve örnek gerilim sinyalleri.

Örnek verileri görüntüle
{
  "kitap_bolumu": 2,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "ruzgar_ve_drone": {
      "kaynak": "bolum02_vektor_geometri.drone_case",
      "coordinate_order": [
        "east",
        "north"
      ],
      "velocity_unit": "m/s",
      "displacement_unit": "m",
      "time_unit": "s",
      "air_velocity_mps": [
        12.0,
        5.0
      ],
      "wind_velocity_mps": [
        -4.0,
        3.0
      ],
      "target_displacement_m": [
        80.0,
        60.0
      ],
      "update_time_s": 10.0
    },
    "gerilim_ornekleri": {
      "kaynak": "bolum02_vektor_geometri.signal_case",
      "sample_unit": "V",
      "reference_v": [
        1.0,
        2.0,
        3.0,
        4.0
      ],
      "observed_v": [
        1.2,
        1.8,
        3.3,
        3.7
      ]
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum02_vektor_geometri.py

Matrisler ve Dönüşümler

Bir LiDAR noktasını sensör, robot ve harita koordinatları arasında taşıma.

Örnek verileri görüntüle
{
  "kitap_bolumu": 3,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "lidar_koordinat_zinciri": {
      "kaynak": "bolum03_matris_donusumleri: SENSOR_* ve BODY_IN_MAP_*",
      "sensor_point_m": [
        2.0,
        1.0
      ],
      "sensor_forward_unit": [
        1.0,
        0.0
      ],
      "sensor_in_body_translation_m": [
        0.5,
        0.2
      ],
      "body_in_map_translation_m": [
        4.0,
        3.0
      ],
      "sensor_in_body_angle_rad": 1.5707963267948966,
      "body_in_map_angle_rad": 0.5235987755982988
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum03_matris_donusumleri.py

Determinantlar, Minörler ve Rank

Tam sayılı Sarrus örnekleri ve determinantla alan ölçeği.

Örnek verileri görüntüle
{
  "kitap_bolumu": 4,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "sarrus": {
      "kaynak": "sarrus_kontrol: example ve parameter_matrix",
      "birim": "boyutsuz",
      "A": [
        [
          2,
          1,
          0
        ],
        [
          3,
          -1,
          2
        ],
        [
          1,
          4,
          5
        ]
      ],
      "t_degerleri": [
        -3,
        -2,
        -1,
        0,
        1,
        2,
        3
      ],
      "G_t": [
        [
          [
            -3,
            1,
            1
          ],
          [
            1,
            -3,
            1
          ],
          [
            1,
            1,
            -3
          ]
        ],
        [
          [
            -2,
            1,
            1
          ],
          [
            1,
            -2,
            1
          ],
          [
            1,
            1,
            -2
          ]
        ],
        [
          [
            -1,
            1,
            1
          ],
          [
            1,
            -1,
            1
          ],
          [
            1,
            1,
            -1
          ]
        ],
        [
          [
            0,
            1,
            1
          ],
          [
            1,
            0,
            1
          ],
          [
            1,
            1,
            0
          ]
        ],
        [
          [
            1,
            1,
            1
          ],
          [
            1,
            1,
            1
          ],
          [
            1,
            1,
            1
          ]
        ],
        [
          [
            2,
            1,
            1
          ],
          [
            1,
            2,
            1
          ],
          [
            1,
            1,
            2
          ]
        ],
        [
          [
            3,
            1,
            1
          ],
          [
            1,
            3,
            1
          ],
          [
            1,
            1,
            3
          ]
        ]
      ]
    },
    "ucgen": {
      "kaynak": "bolum07_koordinatlar_determinant.main_triangle_vertices",
      "vertices_m": [
        [
          100.0,
          50.0
        ],
        [
          104.0,
          51.0
        ],
        [
          101.0,
          53.0
        ]
      ]
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum07_koordinatlar_determinant.py
python3 -B python/sarrus_kontrol.py

Lineer Denklem Sistemleri: Dengeden Çözüme

Malzeme dengesi, direnç ağında düğüm gerilimleri ve Cramer örnekleri.

Örnek verileri görüntüle
{
  "kitap_bolumu": 5,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "malzeme_dengesi": {
      "kaynak": "bolum04_lineer_sistemler.material_balance",
      "akis_ve_fiziksel_sag_taraf_birimi": "kg/h",
      "A_scaled": [
        [
          1.0,
          1.0,
          1.0
        ],
        [
          2.0,
          5.0,
          8.0
        ],
        [
          6.0,
          2.0,
          1.0
        ]
      ],
      "b_scaled": [
        100.0,
        530.0,
        250.0
      ],
      "A_physical": [
        [
          1.0,
          1.0,
          1.0
        ],
        [
          0.2,
          0.5,
          0.8
        ],
        [
          0.6,
          0.2,
          0.1
        ]
      ],
      "b_physical": [
        100.0,
        53.0,
        25.0
      ]
    },
    "direnc_agi": {
      "kaynak": "bolum04_lineer_sistemler.resistor_network",
      "G_birimi": "S",
      "current_birimi": "A",
      "G": [
        [
          4.0,
          -1.0,
          -1.0
        ],
        [
          -1.0,
          3.0,
          -1.0
        ],
        [
          -1.0,
          -1.0,
          3.0
        ]
      ],
      "current": [
        5.0,
        0.0,
        0.0
      ]
    },
    "cramer": {
      "kaynak": "cramer_kontrol.CASES",
      "birim": "boyutsuz",
      "sistemler": [
        {
          "A": [
            [
              2,
              1
            ],
            [
              1,
              -1
            ]
          ],
          "b": [
            5,
            1
          ]
        },
        {
          "A": [
            [
              1,
              1,
              1
            ],
            [
              2,
              -1,
              1
            ],
            [
              1,
              2,
              -1
            ]
          ],
          "b": [
            6,
            3,
            2
          ]
        },
        {
          "A": [
            [
              3,
              -2
            ],
            [
              5,
              4
            ]
          ],
          "b": [
            7,
            1
          ]
        },
        {
          "A": [
            [
              2,
              -1,
              1
            ],
            [
              1,
              2,
              -1
            ],
            [
              3,
              1,
              2
            ]
          ],
          "b": [
            7,
            -6,
            7
          ]
        }
      ]
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum04_lineer_sistemler.py
python3 -B python/cramer_kontrol.py

Yapı, Uzay ve Koordinatlar

Vektör Uzayları, Baz, Boyut ve Baz Değişimi

Rijit plakanın hareket modları ve iki baz arasında koordinat dönüşümü.

Örnek verileri görüntüle
{
  "kitap_bolumu": 6,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "rijit_plaka": {
      "kaynak": "bolum05_vektor_uzaylari.chapter_results.plate_case",
      "measurement_order": [
        "d1x",
        "d1y",
        "d2x",
        "d2y"
      ],
      "displacement_unit": "m",
      "basis_columns": [
        [
          1.0,
          0.0,
          0.0
        ],
        [
          0.0,
          1.0,
          -1.0
        ],
        [
          1.0,
          0.0,
          0.0
        ],
        [
          0.0,
          1.0,
          1.0
        ]
      ],
      "observed_displacement_m": [
        0.01,
        0.02,
        0.01,
        0.04
      ],
      "deformation_candidate_m": [
        0.01,
        0.02,
        0.03,
        0.04
      ]
    },
    "baz_degistirme": {
      "kaynak": "bolum07_koordinatlar_determinant.canonical_bases",
      "birim": "boyutsuz",
      "source_basis_columns": [
        [
          2.0,
          1.0
        ],
        [
          1.0,
          2.0
        ]
      ],
      "target_basis_columns": [
        [
          1.0,
          -1.0
        ],
        [
          1.0,
          1.0
        ]
      ]
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum05_vektor_uzaylari.py
python3 -B python/applications/python/bolum07_koordinatlar_determinant.py

Dört Temel Alt Uzay ve Rank

Eyleyici ve sensör modelinde erişilebilir hedefler ve çekirdek.

Örnek verileri görüntüle
{
  "kitap_bolumu": 7,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "eyleyici_sensor": {
      "kaynak": "bolum06_temel_alt_uzaylar.chapter_results.actuator_sensor_case",
      "matrix": [
        [
          1.0,
          0.0,
          1.0,
          0.0
        ],
        [
          0.0,
          1.0,
          1.0,
          1.0
        ],
        [
          1.0,
          1.0,
          2.0,
          1.0
        ]
      ],
      "input_labels": [
        "u1",
        "u2",
        "u3",
        "u4"
      ],
      "output_labels": [
        "y1",
        "y2",
        "y3"
      ],
      "input_unit": "dimensionless_fraction_of_reference_command",
      "output_unit": "dimensionless_calibrated_feature",
      "reachable_target": [
        2.0,
        3.0,
        5.0
      ],
      "inconsistent_target": [
        2.0,
        3.0,
        5.2
      ],
      "same_output_commands": {
        "first": [
          2.0,
          3.0,
          0.0,
          0.0
        ],
        "second": [
          2.0,
          2.0,
          0.0,
          1.0
        ]
      }
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum06_temel_alt_uzaylar.py

Ortogonallik, İzdüşüm ve QR Ayrışımı

İvme ölçümünü izdüşümle ayrıştırma ve QR ile denetleme.

Örnek verileri görüntüle
{
  "kitap_bolumu": 8,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "ivme_izdusumu": {
      "kaynak": "bolum08_ortogonallik_qr.main_direction_matrix ve main_measurement_vector",
      "direction_matrix_dimensionless": [
        [
          2.0,
          3.0
        ],
        [
          1.0,
          1.0
        ],
        [
          2.0,
          1.0
        ]
      ],
      "acceleration_m_per_s2": [
        1.0,
        2.0,
        -1.0
      ]
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum08_ortogonallik_qr.py

Yaklaştırma, Ayrıştırma ve Güvenilir Hesaplama

En Küçük Kareler ve Kalibrasyon

Yük hücresi kalibrasyonu ve ağırlıklı nivelman hesabı.

Örnek verileri görüntüle
{
  "kitap_bolumu": 9,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "yuk_hucresi": {
      "kaynak": "bolum09_en_kucuk_kareler.load_cell_dataset",
      "force_newton": [
        0.0,
        0.0,
        250.0,
        250.0,
        500.0,
        500.0,
        750.0,
        750.0,
        1000.0,
        1000.0
      ],
      "temperature_change_kelvin": [
        -5.0,
        -5.0,
        5.0,
        5.0,
        0.0,
        0.0,
        -5.0,
        -5.0,
        5.0,
        5.0
      ],
      "response_millivolt": [
        0.16000000000000003,
        0.14,
        1.27,
        1.23,
        2.21,
        2.1900000000000004,
        3.1700000000000004,
        3.1300000000000003,
        4.26,
        4.24
      ],
      "standard_deviation_millivolt": [
        0.015,
        0.015,
        0.025,
        0.025,
        0.02,
        0.02,
        0.03,
        0.03,
        0.02,
        0.02
      ],
      "deterministic_residual_millivolt": [
        0.01,
        -0.01,
        0.02,
        -0.02,
        0.01,
        -0.01,
        0.02,
        -0.02,
        0.01,
        -0.01
      ],
      "true_physical_coefficients": [
        0.2,
        0.004,
        0.01
      ],
      "force_scale_newton": 250.0,
      "temperature_scale_kelvin": 5.0,
      "check": {
        "force_newton": 625.0,
        "temperature_change_kelvin": 2.0,
        "observed_response_millivolt": 2.73
      }
    },
    "nivelman_agi": {
      "kaynak": "bolum09_en_kucuk_kareler.leveling_network_case",
      "design": [
        [
          1.0,
          0.0
        ],
        [
          -1.0,
          1.0
        ],
        [
          0.0,
          1.0
        ]
      ],
      "observed_height_differences_metre": [
        1.02,
        0.98,
        2.01
      ],
      "standard_deviation_metre": [
        0.01,
        0.02,
        0.01
      ],
      "reference_height_a_metre": 100.0
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum09_en_kucuk_kareler.py

Sayısal Doğruluk ve Koşulluluk

Robot yörüngesi ve optik ters problemde ölçek ile koşulluluk.

Örnek verileri görüntüle
{
  "kitap_bolumu": 10,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "robot_yorungesi": {
      "kaynak": "bolum10_sayisal_dogruluk.robot_trajectory_case",
      "start_position_radian": 0.2,
      "end_position_radian": 1.1,
      "duration_seconds": 2.0,
      "boundary_velocity_radian_per_second": [
        0.0,
        0.0
      ],
      "boundary_acceleration_radian_per_second_squared": [
        0.0,
        0.0
      ]
    },
    "optik_ters_problem": {
      "kaynak": "bolum10_sayisal_dogruluk.optical_inverse_problem_case",
      "path_length_centimetre": 1.0,
      "absorbance_unit": "dimensionless",
      "concentration_unit": "mmol/L",
      "sensitivity_entry_unit": "L/mmol",
      "well_separated": {
        "sensitivity_matrix": [
          [
            1.0,
            0.2
          ],
          [
            0.15,
            1.1
          ]
        ],
        "true_concentration_millimole_per_litre": [
          0.8,
          0.4
        ],
        "absorbance_perturbation": [
          0.0001,
          -0.0001
        ]
      },
      "nearly_parallel": {
        "sensitivity_matrix": [
          [
            1.0,
            0.99
          ],
          [
            0.98,
            0.9703
          ]
        ],
        "true_concentration_millimole_per_litre": [
          0.8,
          0.4
        ],
        "absorbance_perturbation": [
          0.0001,
          -0.0001
        ]
      }
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum10_sayisal_dogruluk.py

Simetri, Enerji ve Pozitif Tanımlılık

Mesnetli yay ağında rijitlik, denge ve enerji hesabı.

Örnek verileri görüntüle
{
  "kitap_bolumu": 11,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "mesnetli_yay": {
      "kaynak": "bolum11_simetri_enerji.grounded_spring_case",
      "incidence": [
        [
          1.0,
          0.0,
          0.0
        ],
        [
          -1.0,
          1.0,
          0.0
        ],
        [
          0.0,
          -1.0,
          1.0
        ],
        [
          0.0,
          0.0,
          1.0
        ]
      ],
      "spring_constants": [
        1200.0,
        800.0,
        1500.0,
        1000.0
      ],
      "applied_force": [
        100.0,
        -40.0,
        60.0
      ],
      "perturbation": [
        0.001,
        -0.0005,
        0.0008
      ],
      "units": {
        "spring_constant": "N/m",
        "applied_force": "N",
        "displacement": "m"
      }
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum11_simetri_enerji.py

Özdeğerler, Modlar ve Kararlılık

Üç platformlu yapının titreşim modları ve kararlılık örnekleri.

Örnek verileri görüntüle
{
  "kitap_bolumu": 12,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "uc_platform": {
      "kaynak": "bolum12_ozdeger_modlar.three_platform_modal_case",
      "masses_kg": [
        2000.0,
        1500.0,
        1000.0
      ],
      "story_stiffnesses_n_per_m": [
        8000000.0,
        6000000.0,
        4000000.0
      ],
      "assumptions": [
        "small horizontal displacements",
        "linear elastic story springs",
        "lumped platform masses",
        "fixed base",
        "undamped free vibration for modal extraction",
        "constant time-invariant parameters"
      ]
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum12_ozdeger_modlar.py

Tekil Değer Ayrıştırması, Düşük Rank ve Veri Yapısı

Köprü sensörleri ve sentetik termal görüntüde düşük rank yaklaşımı.

Örnek verileri görüntüle
{
  "kitap_bolumu": 13,
  "tur": "Öğretim örneği girdileri",
  "ornekler": {
    "kopru_sensoru": {
      "kaynak": "bolum13_svd_dusuk_rank.bridge_strain_sensor_data",
      "observation_indices": [
        1,
        2,
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        ]
      ]
    }
  }
}
Çalıştırma komutları
python3 -B python/applications/python/bolum13_svd_dusuk_rank.py