Linear Algebra Flashcards

Invertible Matrix Properties

• A matrix A is defined as invertible when there exists another matrix, denoted as C, such that the product CA equals the identity matrix I.
• The matrix must also be row equivalent to the n x n identity matrix, indicating full rank.

Pivots and Solutions

• An invertible matrix A contains n pivots, which are the leading coefficients in its row echelon form that indicate linearly independent rows.
• The equation ax=0 has only the trivial solution (x = 0), confirming the linear independence of columns.

Linear Independence and Transformation

• The columns of A form a linearly independent set, meaning no column can be expressed as a linear combination of others.
• The linear transformation defined as x -> Ax is one-to-one, implying distinct inputs map to distinct outputs.

Span and Basis

• For every vector b in R^n, the equation ax = b has at least one solution, meaning the columns of A span the entirety of R^n.
• The transformation x -> ax maps Rn onto Rn, signifying that every output is achievable from some input.

Basis, Column Space, and Dimensions

• The columns of A not only span R^n but also form a basis, indicating that they are both spanning and linearly independent.
• The column space of A, denoted as Col A, equals R^n, confirming that the set completely fills the n-dimensional space.

Dimensionality and Rank

• The dimension of the column space is given by dim Col A = n, representing its maximal dimensionality.
• The rank of the matrix A is n, indicating the maximum number of linearly independent column vectors it contains.

Null Space Properties

• The null space of A, denoted as Nul A, contains only the zero vector, indicating no non-trivial solutions exist for the homogeneous system.
• The dimension of the null space is dim Nul A = 0, reinforcing the idea that A is invertible as it lacks free variables.