Matrices
-n x m matrix
-row vector
-column vector
-adding matrices
-multiplying matrices
-scalar multiplication of matrices
-square matrix
Systems of Linear Equations
- writing systems of equations as matrix multiplication
- a particular solution vs the solution set
- consistent system of equations (has at least one solution)
- inconsistent system of equations (no solution)
- geometric meaning
Geometry
- vector equation of a line (one parameter)
- (new) vector equation of a plane (two parameters)
- vector equation of a hyperplane (more than 2 parameters)

Solving systems of equations
- augmented matrix
- row operations (on a matrix)

- equivalent systems of equations (book also calls this "row equivalent") - row echelon form
- Gaussian Elimination
- reduced row echelon form
- Gauss-Jordan Elimination

Matrix Equations - Solving matrix equations AX=B
- "augmented matrix" of a matrix equation: (A|B)
- Identity matrix: I
- Inverse matrix

- row equivalent matrices
- coefficient matrix
- rank of a matrix (rk(A))
- relation of rank to solutions of equations
- homogeneous system of equations (Ax=0)
- associated homogeneous system of equations ( Ax=B has associated system Ax=0 )

Determinants and More

- Using cofactor expansion to calculate determinants
- diagonal matrix
- upper triangular matrix
- lower triangular matrix
- transpose of a matrix
- elementary matrices
- how row operations affect determinants

- relationship between rk(A), A inverse, det(A), RRE form of A, solutions to Ax=0
- If E is an elementary matrix then EA is the matrix A with the row operation of E done to it

- det(AB)=det(A)det(B)

- adjoint of a matrix
Vector Spaces

-linear combination
-span{v1,v2,...,vn}
-linearly dependent set of vectors
-linearly independent set of vectors
-subspace
-spanning set (for a subspace)
-minimal spanning set
-basis

Eigenvalues and Eigenvectors

- eigenvalue of a matrix A (a number r such that Ax=rx)
- eigenvector (associated to an eigenvalue r -> non zero vector x such that Ax=rx)
- eigenspace (associated to eigenvalue r -> ker(rI-A))
- characteristic polynomial (of A -> p(r)=det(rI-A) )
- algebraic multiplicity (of an eigenvalue)
- geometric multiplicity (of an eigenvalue)
- diagonalizable

Applications

- Leontief Model (closed and open)
- Difference Equations (ie The rabbit stuff in 4.2)

Linear Transformations

- T:R^n-->R^m is called a linear transformation if T(cv)=cT(v) and T(v+w)=T(v)+T(w)
- T is always given by T(v)=Av where A is an m x n matrix
- A is called the standard matrix of T
- Given any basis {v1,...,vn} of R^n and any set of n vectors {w1,...wn} of R^m we can define a linear transformation T:R^n --> R^m by setting T(v1)=w1, T(v2)=w2 ... T(vn)=wn.
- im(T) the image of T (=COL(A))
- ker(T) the kernel of T (=ker(A))
- rk(T) is the dimension of im(T)
- null(T) is the dimension of ker(T)
- Dimension Theorem: rk(T)+null(T)=n
- T is onto if im(T)=R^m (ie rk(T)=m)
- T is one to one if whenever T(v)=T(w) then v=w
- T is one to one if and only if ker(T)={0} (ie null(T)=0)
- T: R^n --> R^n is an isomorphism if it is onto and one to one. (by the Dimension Theorem we only need to check one of these)
- standard matrix of a rotation, reflection and projection in R^2
- composition of maps: T:R^n--> R^m, T':R^m-->R^s then T'T:R^n-->R^s and T'T has standard matrix A'A.