Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.
Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.
Equivalence theorem for nonsingular matrices: the null space of the matrix A is {0}.
Equivalence theorem for nonsingular matrices: there is a pivot position in every row of A.
Equivalence theorem for nonsingular matrices: the rows of A are a basis for R^n (or C^n).
Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.
Equivalence theorem for nonsingular matrices: the rows of A span R^n (or C^n).
Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.
Equivalence theorems for injective transformations
Equivalent matrices represent the same linear transformation with resect to appropriate bases.
Every basis for a vector space contains the same number of elements
Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.
Every matrix has an eigenvalue over the complex numbers.
Every matrix is row-equivalent to a matrix in reduced row echelon form.
Every matrix is row-equivalent to only one matrix in reduced row echelon form.
Every nilpotent matrix is similar to one with 1 on subdiagonal blocks and all other entries 0.
Every square matrix is conjugate
Every square matrix is similar the sum of a diagonal and a nilpotent matrix.
Every square matrix is similar to one in Jordan form.
Example of a linear transformation on R^2: generic
Example of a linear transformation on R^2: projection
Example of a linear transformation on R^2: rotation
Example of a linear transformation on R^2: shear
Example of a linear transformation on R^3: rotation
Example of a sum of vectors interpreted geometrically in R^2

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