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What is the rank of an m x n matrix A?
the dimension of the column space
What is the column space of an m x n matrix A?
the set of all linear combinations of the columns of A
What is the nullity of A?
The dimension of the null space
What is the null space of an m x n matrix A?
The set of all solutions of the homogeneous equation Ax = 0
What does Ax = 0 mean?
A: coefficient matrix
x: is the vector of variables
0: the zero vector
What is the Rank Theorem?
rank A + nullity A = number of columns in A
OR
dim(r(t)) + dim(ker(t)) = dim(dom)
What is one-to-one?
A mapping T: Rn => Rm is said to be one-to-one if each b in Rm is the image of AT MOST ONE x in Rn
What is onto?
A mapping T: Rn => Rm is said to be onto Rm if each b in Rm is the image of AT LEAST ONE x in Rn