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# Linear Algebra 4.1-2

### Quiz

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

Eight rules for a linear space or vector space | associative, commutative, neutral element,has negative, distributive, associative with products, same as mulitple of 1 |

A linear space is | a set with two reasonably defined operations, addition and scalar multiplication, that allow us to form linear combinations. |

The neutral element for an nxm matrix | the 0 matrix |

The neutral elemtn for the linear equations in three unknowns: ax+by+cz=d | is a=b=c=d=0 |

The neutral element of a vector | the 0 vector |

K(a+ib)=ka+i(kb) form a linear space and have the neutral element | 0=0+0i |

Subspaces: A subset W of a linear space V is called a subspace of V if | a. contains the neutral element 0 of V b. is closed under addition c. W is closed under scalar multiplication (b/c is closed under linear combinations) |

Differentiable functions form a subspace, True or False? | True |

Examples of subspaces | c^(infinity), P the set of all polynomials, Pn the set of all polynomials of decree less than or equal to n |

we say that f1....f2 span V if | every f in V can be expressed as a linear combination of f1...f2 |

We say that fi is redundant if | it is a linear combinationf f1...fi-1. The elements are called linearly independent if none of them is redundant. |

If equation c1f1+...+cnfn=0 | has only the trivial solution c1=...=cn=0 |

We say that f1...fn are a basis of V if they | span V and are linearly independent. This means that every f in V can be written uniquely as a linear combination. |

The coefficients c1...cn are called the | coordinates of f with respect to the basis B=(f1...fn) |

B-coordinate transformation is invertible 1 | True |

[f+g]B=[g]B+[f]B T/F | T |

Finding a basis of a linear space | a. Write down a typical element of V, int terms of some arbitrary constants b. Use arbitrary constants as coeff, express typical element as a linear combination of some elements of V c. Verify the elements are linearly independent |

Finite dimensional linear spaces | A linear space V is called finite dimensional if it has a (finite)basis so that we can define its demenension dim(V)=n |

The image is a subspace of the and the kernel is a subspace of the | target-domain |

If the image of T is finite dimensional, then dim(imT) is called the | rank of T |

If the kernel of T is finite dimensional, then dim(kerT) is called | nullity of T |

dim(V)= | rank(T) +nullity(T) |

dim(V)= | dim(imT)+dim(kerT) |

Show linear transformation | show that closed under scalar multiplication and addition |

The kernel consists of | all inputs into T that produce 0 as output |

Is T 1-1 check | Ker(T) must =0 |

2out of 3 Rule | -dim(V)=dim(W) -ker(T)=0 (one-to-one) -im(T)=W (onto) |

If one of the three is false then | Not isomorphic |

If 2 out of three is true | Isomporphic |

P2-> LB and LC-> R^3 | |

Sc->B = | (SB->C)^-1 |

Isomorphic | same structure |

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ok2bpure