2.1, 2.2, 2.3 and Rank (4.6)

THEOREM Transpose Matrix

1) (A^T)^T=A 2)(A+B)^T=A^T+B^T 3)(rA)^T=r(A^T) *4) (AB)^T=B^TA^T THIS ONE IS IMPORTANT

Algorithm for finding Invertible Matrix of A

1) Input a n*n matrix A 2) Write [A:I] 3) Row reduce A to the reduce echelon form, If that produces the identity matrix then this produces [I:A^-1] OTHERWISE, A is not invertible If there is a row full of zeros in the left side A is not invertible

Properties of Matrix Multiplication

1) It is possible for A*B to exist but for B*A not to be possible. 2) AB=AC does not imply B=C 3) AB=0 does not imply A 0r B=0

Types of Isomorphisms

1) P<=r -->R^(r+1) 2)M(m*n)-->R^(mn)

Isomorphism

A linear transformation L:R^n->R^n is called invertible or an isomorphism if there exists a linear transformation S:Rn->Rn such that SL=id and LS=id

Square Matrix

A matrix is a square matrix if the number of rows is equal to the number of columns If A is n*n then, A^s=A*A*A*A*A.... s times A^0 is the identity matrix

THEOREM: Invertibility of a Matrix

A n*n matrix is invertible if and only if it can be row reduced to the identity matrix (In). In this case, the same elementary row operations produce A^-1 when applied to (In). See theorem 8 in section 2.3

Elementary Matrix

A square matrix is called elementary if it can be obtained from the identity matrix of the same size by one elementary row operation An elementary matrix E is invertible and its inverse E^-1 is obtained from I by performing the inverse of the elementary row operations that produced E. See example

Identity Matrix

A*I(m)=A I(n)*A=A whats the difference between these two?

Inverse Matrix

A^-1 A*A^-1=I A^-1*A=I

Transpose Matrix

A^T has columns that are the corresponding rows of A

Invertible Matrix

An n*n matrix is called invertible if there exists an n*n matrix C such that CA=In AC=In -In this case we say that C is an inverse of A -SHOW THAT THE INVERSE OF A IS UNIQUE

Inverse Matrix and System of Linear Equations

Ax=b A^-1*Ax=A^-1*b Ix=(A^-1)b x=(A^-1)b If A is invertible, then Ax=b has a unique solution x=(A^-1)b

Elementary Matrices as Elementary Row Operations

Elementary row operations can be performed by multiplying by elementary matrices (E1)A ->first elementary operation (E2)(E2)A ->second elementary operation (Ep).....(E1)A is the Reduced Echelon Form of A

THEOREM: The Rank-Nullity Theorem

Let A be a m*n matrix, then rankA+dimNulA=n dimNulA=Ker(A)

THE INVERTIBLE MATRIX THEOREM

Let A be a square n*n matrix. The following are logically equivalent 1) A is an invertible matrix 2) A is row equivalent to the n*n identity matrix 3) A has n pivot positions 4) The equation Ax=0 has only the trivial solution 5) The columns of A form a linearly independent set 6) The linear transformation x-->Ax is one-to-one 7) the equation Ax=b has at least one solution for every b in Rn 8) The columns of A span Rn 9) The linear transformation x-->Ax maps Rn onto Rn 10) There is an n*n matrix C such that CA=I 11) There is an n*n matrix D such that AD=I 12) A^T is an invertible matrix 13) the columns of A form a basis of Rn 14) ColA=Rn 15) dimCola=n 16) rankA=n 17) NulA={0} 18) dimNulA=0 19) the number 0 is NOT an eigenvalue of A 20) the determinant of A is not 0

THEOREM of Matrix Multiplication

Let A be an m*n matrix and B,C be appropriate size to make the sums and products below well defined 1) A(BC)=(AB)C 2) A(B+C)=AB+AC (B+C)A=BA+CA 3) r(AB)=(rA)B=A(rB) for all r real numbers

Rank Corollary

Let A be an n*n matrix TFAE 1) A is invertible 2)rankA=n 3)ColA=Rn 4)RowA=Rn 5)NulA={0}

Inverse of A and Elementary Matrices

Let Ep*....*E1*A=I (Ep....E2E1)^-1(Ep....E2E1)A=(Ep...E2E1)^-1 multiple each side by the inverse of the elementary matrices A=(Ep...E2E1)^-1 A^-1=Ep*...*E2E1

THEOREM: Invertible Transformations

Let L: R^n-->R^n be a linear transformation. Let A be the standard matrix of L. L is invertible (isomorphism) if and only if A is (invertible?) WHHHHHAAATTTT!!!! in this case, the linear transformation S with standard matrix A^-1 satisfies S(0)L=id and L(0)S=id

THEOREM: Characteristics of Invertible Matrices

Let and B be invertible n*n matrices, Then 1) A^-1 is invertible and (A^-1)^-1=A 2) AB is invertible and (AB)^-1=B^-1(A^-1) HAVE TO SWITCH ORDER BECAUSE TRANSPOSED 3) A^T is invertible and (A^T)^-1=(A^-1)^T

Matrix Multiplication

Not Commutative... A*B is not the same as B*A

Row Space

The rows of A span RowA

THEOREM: RANK of A

dim (RowA)=dim(ColA) this number is called the rank of A compute rank by row reducing a matrix and counting pivot columns