Linear Algebra: The Unifying Theorem

Study Notes

The Unifying Theorem

Let S be a set of n vectors in R^n, and A be a matrix with transformation T: R^n -> R^n defined by T(x) = Ax.
Transformation T is both onto (surjective) and one-to-one (injective), indicating a bijective relationship.
Set S spans R^n, confirming that every vector in R^n can be expressed as a linear combination of vectors in S.
Set S is linearly independent, establishing that no vector in S can be formed from others in the set.
Set S acts as a basis for R^n, fulfilling both spanning and independence criteria.
The equation Ax = b has a unique solution, affirming that every b in R^n corresponds to exactly one x.
Matrix A is invertible, which signifies that it has an inverse A^(-1).
The kernel of T (ker(T)) equals the null space of A (null(A)), and is {0}, confirming that the only solution to Ax = 0 is the trivial solution.
Nullity of A is 0, indicating there are no free variables in the system.
The column space (col(A)) equals the range of A, which is R^n, confirming full rank.
The row space (row(A)) is also R^n, indicating the rows of A span the space.
The rank of matrix A is n, meaning it has full rank and all n rows/columns are linearly independent.
The determinant of A is non-zero (det(A) != 0), confirming that the matrix is full rank and invertible.
Eigenvalue condition: λ = 0 is not an eigenvalue of matrix A, indicating no solutions for Ax = 0 beyond the trivial solution.

Description

This quiz explores the Unifying Theorem in Linear Algebra, detailing the properties of the transformation T: R^n -> R^n defined by the matrix A. Participants will examine concepts such as linear independence, spanning sets, and the unique solvability of equations in R^n. Test your understanding of these core principles and their implications for vector spaces.