Math Test 2

3 terms a subset must satisfy

1. 0 e W

2. closed under addition: if u e w then u+v e W

3. clsoed under scalar mulitplication if u e w, c e R, then cu e W

T or F: Every span is a subspace

T

Def of a Bases

• a set of vectors is a basis for a subspace if B spans H and B is linearly independent

• given a set of vectors to find the basis remove vectors that cause the set to NOT be linearly independent so it becomes linearly independent

Nul(A) (def, lives in and membership test)

• Ax = 0

• R^n

• to test a specific u, compute Au

Col(A) (def, lives in and membership test)

• spans of the comuns of A

• R^m

• b e Col(A) when Ax = b is consistent

Row(A) (def, lives in and membership test)

• span of the rows of A or Col(A^T)

• R^n

• rr and compare with a basis of row space

how to find basis for Nul(A)

1. row reduce

2. solve for Ax = 0

3. write the sol in parametric vector form

4. vectors mulitple by the free variables from the basis

how to find basis for Col(A)

• row reduce A to find pivot columns

• select the corresponding pivot columns from the original matrix A

how to find basis for Row(A)

• row reduce A to echelon form

• the non zero rows form a basis

2 formula for basis

x = Pb[x]b

[x]b = Pb^-1*x

rank(A)

• number of pivots

• 
  

nullity(A)

• number of free varibales

• rank + nullity = number of columns

The basis thrm

any linearly independent set of numbers is automatically a basis

3 parts of a them that is how row operations affect determinats

1. multiple of 1 row added to another detB = detA

2. rows interchanged detB = - detA

3. a row is mulitpled by k then detB = kdetA

Is the matrix invertiable if the det = 0

No, the det must NOT equal 0 to be invertible

methods to find the det of a matrix (3)

• cofactor expansion

  • choose row/colm to delete using Cij = (-1)^(i+j) * det(Aij) (pick the one with the most zeros)

• if a triangluar matrix then the det(A) = product of the entries on the diagonal

crammers rule

xi = det(Ai(b))/det(A)

gives you the solution to a system

long version to find a determinant formula

a11a22a33 + a21a32a13 + a31a12a23 - a13a22a31 - a11a23a32 - a12a21a33

what is the adjugate matrix

The adjugate matrix is the transpose of the matrix of cofactors. [For instance, C12 goes in the (2, 1) position

how to know if W spans H?

vector \(\mathbf v\) spans a set \(W\), you check whether every vector in \(W\) can be written as a scalar multiple of \(\mathbf v\)

T or F: the span of any set of vectors in R³ is a subspace of R³

T

dimension of the spanned subspace is equal to

rank

def of a consistent matrix

there is NOT a pivot in the last colm

dim (Row (A) =

rank(A)

dim(nul(a)) =

nuliity = number of free variables

dimension of the space spanned by a set of vectors

equal to the rank = number of pivot columns