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# Linear Transformatio

### Linear Transformation, Kernel, Range, Eigenvalue/Vector, diagonalization

Question | Answer |
---|---|

Linear Transformation | Let V & W be two vectors spaces. A map T frm V to W. |

Linear Transformation from Rn to Rm is of the form ... | T(x) = Ax with A= [T(e1):T(e2):...:T(en)] |

Ck(I) | Space of continuous function on the I interval |

Mapping T:(V -W) with V and W be vector spaces | T from V onto W is a rule that assigns to each vector v in V precisely one vector w=T(v). |

Linearity properties | T(u+v) = T(u) = T(v) for all u,v in V T(cv) = cT(v) for all v in V and all scalars c. |

A mapping T:V-W is called ilnear transforamtion from V to W if .... | Linear Properties are true. The vector space V is called the domain of T, whie the vector space W is called the codomain of T. |

Let T:V-W be a linear transformation. Then ... | T(0v) = 0w T(-v) = -T(v) for all v in V. |

A linear transformation T:Rn - Rm defined by T(x) = Ax is an m x n matrix called ...; | Matrix Transformation |

Ker (T) = | {v in V: Tv = 0} |

Ran ( T) = (or Rng(T)) | {T(v) of W : v in V} |

Kernel | Let T:V-W be a linear transforamtion.the set of all vector v in V such that T(v)=0 |

Range | A linear transformation T:V-W , the subset of W consisting of all transformed vector from V. |

If T:Rn-Rm is the linear transformation with matrix A, then ... | Ket(T) is the solution set to the homogeneous linear system Ax=0. |

If T:Rn - Rm with m x n matrix A , then Ket(T) = | nullspace(A) subspace of Rn |

If T:Rn - Rm with m x n matrix A , then Ran(T) = | colspace (A) subspace of Rm |

If T:V-W is a linear transformation, then Ker(T) = | is a subspace of V |

If T:V-W is a linear transformation, then Ran(T) = | subspace of W |

General Rank-nullity theorem | dim[Ran(T)] + dim[Ker(T)] = dim[V] |

Created by:
DrMolina