Math 220 - Matrix Theory and Linear Algebra

These flashcards cover essential vocabulary and concepts from Math 220 relating to linear algebra, matrix operations, and systems of equations.

Gaussian Elimination

A method for solving linear systems by reducing them to row echelon form and then finding the solutions.

Augmented Matrix

A matrix that includes the coefficients and the constants of a linear system.

Row Echelon Form

A form of a matrix where all non-zero rows are above any rows of all zeros and leading coefficients of a row are to the right of the leading coefficients of the previous row.

Infinite Solutions

A scenario in a system of equations where there are more variables than independent equations, leading to infinite solutions.

Unique Solution

When a system of equations has exactly one solution, often indicated by a row of the augmented matrix signifying a variable in terms of constants.

No Solution

Occurs when the system of equations is inconsistent, meaning that there is no set of variable values that satisfies all equations.

Linear Combination

A mathematical combination of vectors where coefficients are applied to each vector and summed together.

Span

The set of all possible linear combinations of a given set of vectors; forms a vector space.

Basis of a Vector Space

A set of vectors in a vector space that are linearly independent and span the entire vector space.

Dimension of a Vector Space

The number of vectors in a basis for the vector space, indicating the number of dimensions it occupies.

Eigenvalue

A scalar associated with a linear transformation represented by a matrix, indicating how much the eigenvector is stretched or compressed.

Eigenvector

A non-zero vector that changes only by a scale factor when a linear transformation is applied.

Singular Matrix

A square matrix that does not have an inverse, typically indicated by a determinant of zero.

Rank of a Matrix

The dimension of the vector space generated by the rows or columns of the matrix; indicates the maximum number of linearly independent row or column vectors.

Null Space

The set of all vectors that result in the zero vector when multiplied by a given matrix.

Determinant

A scalar value derived from a square matrix that provides information about the whether matrix is invertible.

Orthogonal Vectors

Vectors that are perpendicular to each other in a vector space, with a dot product of zero.

Linear Transformation

A mathematical function between two vector spaces that preserves vector addition and scalar multiplication.

Homogeneous system

A system of linear equations that all equal zero

Augmented Matrix Notation

A notation used to represent a system of linear equations, combining the coefficients of the variables and the constants into a single matrix.

Coefficient Matrix

A matrix consisting solely of the coefficients of the variables in a system of linear equations.

transpose

A matrix where the row become the columns

Domain

euclidian space of inputs?

Codomain

Euclidian space of outputs

Image

The output of a specific vector

Range

The collections of all images(or all possible outputs)

Upper triangular matrix

When all the entries below the main diagonal or zero

lower triangular matrix

When all the entries above the main diagonal are zero

A diagonal matrix

When the entries above and below the main diagonal or zero

Subset

A collection of some of the elements from the original set

Subspace

Rank

The dimensions of the column space

Nullspace

A subset of all vectors V where AV=0

Column space

The band of the columns in a matrix

Nullity