Definition: A transformation from Rn to Rm is a rule T that assigns each vector x in Rn a vector T(x) in Rm, defined by T(x) = Ax
For x in Rn, the vector T(x) is the image of x under T, corresponds
The set of all images is the range of T
3.2 One to One and Onto Functions
One to one: for every vector in Rn(input) there is a unique vector in Rm (image, output)
If T(u) = T(v), u=v
It means matrix A has only trivial solution and pivot every column
Onto: for every vector in Rn(input) there is at least 1 vector in Rm (image, output)
Matrix A has pivots every row
Linear transformation T: Rn to Rm satisfying:
3.5 Matrix Vertices
Let A be nxn matrix, we say A is invertible if there is an nxn matrix B such that AB = In BA = In
B is the inverse of A
A-1 is invertible and its inverse is A
AB is invertible and its inverse is (AB)-1 = B-1A-1
Calculating the inverse matrix:
Special for 2x2: Let A = (ab)(cd), then the inverse is (1/det(a))(d-b)(-ca)
If det = 0, then it is not invertible
For traditional one, have to use identity matrix
