Matrix Algebra Flashcards

Echelon Form

• A matrix is in echelon form if all nonzero rows are positioned above rows of all zeros.
• Each leading entry in a row must be to the right of the leading entry in the row above.
• All entries below a leading entry must be zeros.

Reduced Row Echelon Form (RREF)

• Leading entries in each nonzero row are equal to 1.
• Each leading 1 is the only nonzero entry in its respective column.

Pivot Columns

• A column containing a leading 1 in the reduced echelon form, indicating a pivot position.

Basic and Free Variables

• Basic variables correspond to pivot columns in a matrix.
• Free variables are those that do not correspond to a pivot column.

Span

• Span{v1...vp} represents the set of all linear combinations of vectors v1 to vp in R^n, expressed as c1v1 + c2v2 + ... + cpvp.

Product of Matrices (AX)

• The product of matrix A and vector x is a linear combination of the columns of A, with corresponding entries of x serving as weights.

Linear Independence

• A set of vectors is linearly independent if the equation x1v1 + ... + xpvp = 0 has only the trivial solution.
• No vector in the set can be expressed as a linear combination of the others.

Linear Dependence

• A set of vectors is linearly dependent if there exists a non-trivial combination that equals zero.
• A zero vector in the set indicates linear dependence, as does having more variables than equations.

Linear Transformations

• A transformation T is linear if it satisfies T(U+V) = T(U) + T(V) and T(cU) = cT(U).
• A linear transformation from R^n to R^m is onto if every b in R^m can be produced from at least one x in R^n.
• A transformation is one-to-one if it maps each b in R^m to at most one x in R^n, ensuring no free variables and linear independence of columns.

Row-Column Rule for Matrix Multiplication

• If AB is defined, the entry in row i and column j is the sum of products of row i of A and column j of B.

Invertibility of a Linear Transformation

• A transformation T: R^n to R^n is invertible if there exists an inverse function S such that S(T(x)) = x and T(S(x)) = x.

Elementary Matrices

• Elementary matrices are obtained by performing a single elementary row operation on an identity matrix.

Transpose of a Matrix

• The transpose of an mxn matrix A, denoted A^T, is an nxm matrix, where rows of A become columns of A^T.

Homogeneous Equations

• The homogeneous equation Ax = 0 has a nontrivial solution if at least one free variable exists.

Linear Transformation Conditions

• A linear transformation T maps R^n onto R^m if the columns of its standard matrix A span R^m.
• T is one-to-one if the columns of A are linearly independent.

Inverse Matrix Theorem

• A matrix A is invertible if it is row equivalent to the identity matrix, has n pivot positions, and the equation Ax = 0 has only the trivial solution.
• Its columns form a linearly independent set and span R^n.

Inversion and Linear Transformations

• For an invertible n x n matrix A, there is a unique solution for each b in R^n, maintaining consistency across transformations.