| x₁  y₁  1 |

Area= ±1/2 det | x₂  y₂  1 |

               | x₃  y₃  1 |

               | x₁  y₁  1 |

           det | x₂  y₂  1 | = 0

               | x₃  y₃  1 |

Find the equation of a line

passing through two points (R²)

               | x   y   1 |              

           det | x₁  y₁  1 | = 0

               | x₂  y₂  1 |

                  | x₁  y₁  z₁  1 |

                  | x₂  y₂  z₂  1 |

        ± 1/6 det | x₃  y₃  z₃  1 |

                  | x₄  y₄  z₄  1 |

            | x   y   z   1 |

        det | x₁  y₁  z₁  1 | = 0

            | x₂  y₂  z₂  1 |

            | x₃  y₃  z₃  1 |

1.) Closure under addition: u+v is in V

2.) Commutative Property: u+v = v+u

3.) Associative Property: (u+v)+w = u+(v+w)

4.) Zero vector exists: u+0 = u

5.) Additive inverse: u+(-u) = 0

6.) Closure under scalar multiplication: cu is in V

7.) Distributive Property: c(u+v) = cu+cv

8.) Distributive Property: (c+d)u = cu+du

9.) Associative Property: c(du) = (cd)u

10.) Scalar Identity: 1(u) = u

1) form a homogenous system of equations with vectors in columns.

2) RREF

if the det is not 0,

and you find the identity matrix,

if 2 vectors, and one row is not a multiple of another,
the only trivial solution is 0 = Linearly independent

if you find:a free variable or a zero row or,

if one vector can be written as the combination of the others or,

if one row is a multiple of another,

Linearly dependent

If every vector in V can be written as a

linear combination of the given set, the set spans.

(u)=u₁(1,0,0)+u₂(0,1,0)+u₃(0,0,1) = (u_1,u_2,u₃)

-------

1) make a coefficient matrix with vectors as columns

2) if det[S] ≠ 0 , it spans

3) if RREF[S] is the id matrix, it spans

1) is it non-empty?

2) Closure under addition (in set) ?

3) Closure under multiplication (in set) ?

4) does it contain the zero vector [0] ?

If yes to all, it's a subspace.

Does S span V?

Is S linearly independent?

yes to both = basis

(if more vectors than dimension, it's
linearly dependent, therefore not a basis.)

T or F

A Vector space consists of 4 entities: A set of vectors, a set of scalars and two operations.

T or F

The set of all integers with the standard operations is a vector space.

False

(fails closure under multiplication)

1/c

T or F

The set of all pairs of real numbers of the form (x,y) where y ≥ 0 is a vector space.

False

• Fails additive inverse 
• Fails scalar multiplication: -c(x,y) , y is negative.
  

T or F

Every vector space V contains at least one subspace that is the zero subspace.

T or F

If V and W are both subspaces of a vector space U, then the intersection of V and W is a subspace

True

(but not the union.)

T or F

If U, V, and W are vector spaces such that W is a subspace of V and U is a subspace of V, then W = U

T or F

A set of vectors S={v₁, v₂,...,v_n} is
Linearly dependent

if the vector equation c₁v₁,...,c_nv_n = 0

has only the trivial solution.

T or F

Two vectors u and v in a vector space V are

linearly dependent iff one is a
scalar multiple of another.

T or F

If dim(V) = n then there exists a set of n-1 vectors in V that will span V

False

(n will span V)

T or F

If dim(V) = n then there exists a set of n+1 vectors in V that will span V

True

a basis + a random vector is OK, it's

linearly dependent, but counts as a span.

T or F

A set of vectors S={v₁, v₂,...,v_n} is
Linearly independent

if and only if  0v₁+0v_2,...+0v_n = 0

False,

(That's true for every vector.)

T or F

If V is a vector space of dimension n then any set of n+1 vectors is linearly dependent.

T or F

In  ℝ²,  a set of one vector is linearly dependent.

T or F

In  ℝ²,  a set of one vector is linearly independent.

True

(unless it's the zero vector)

T or F

(0,0,0) is linearly dependent.

T or F

The standard operations in Rⁿ are
vector addition and scalar multiplication.

T or F

The additive inverse of a vector is unique

T or F

The additive identity of a vector is not unique