A rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
What is the determinant of a matrix?
A scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation defined by the matrix.
What is the rank of a matrix?
The dimension of the vector space generated by its columns.
What is an eigenvector?
A nonzero vector that, when multiplied by a given matrix, remains proportional to the original vector, i.e., any scalar multiple of the eigenvector is also an eigenvector.
What is a system of linear equations?
A set of linear equations that must be solved together.
What is the inverse of a matrix?
A matrix such that when it is multiplied by the original matrix, the identity matrix is obtained.
What is the transpose of a matrix?
A matrix obtained by interchanging its rows and columns.
What is a vector space?
A collection of vectors that can be added together and multiplied (scaled) by numbers, called scalars.
What is a linear transformation?
A function that maps each vector in a vector space to another vector in the same vector space, that preserves the addition of vectors and scalar multiplication.
What is the dot product of two vectors?
A scalar obtained by taking the sum of the products of the corresponding entries of two vectors.
What is a basis of a vector space?
A set of linearly independent vectors that span the vector space, i.e., any vector in the space can be written as a linear combination of the basis vectors.
What is a linearly independent set?
A set of vectors in which none of the vectors can be written as a linear combination of the others.
What is a linearly dependent set?
A set of vectors in which one or more of the vectors can be written as a linear combination of the others.
What is the null space (kernel) of a matrix?
The set of all solutions to the homogeneous system of linear equations Ax = 0, where A is the matrix.
What is the image (range) of a matrix?
The set of all possible outputs of the linear transformation defined by the matrix.
What is a matrix?
A rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
What is the determinant of a matrix?
A scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation defined by the matrix.
What is the rank of a matrix?
The dimension of the vector space generated by its columns.
What is an eigenvector?
A nonzero vector that, when multiplied by a given matrix, remains proportional to the original vector, i.e., any scalar multiple of the eigenvector is also an eigenvector.
What is a system of linear equations?
A set of linear equations that must be solved together.
What is the inverse of a matrix?
A matrix such that when it is multiplied by the original matrix, the identity matrix is obtained.
What is the transpose of a matrix?
A matrix obtained by interchanging its rows and columns.
What is a vector space?
A collection of vectors that can be added together and multiplied (scaled) by numbers, called scalars.
What is a linear transformation?
A function that maps each vector in a vector space to another vector in the same vector space, that preserves the addition of vectors and scalar multiplication.
What is the dot product of two vectors?
A scalar obtained by taking the sum of the products of the corresponding entries of two vectors.
What is a basis of a vector space?
A set of linearly independent vectors that span the vector space, i.e., any vector in the space can be written as a linear combination of the basis vectors.
What is a linearly independent set?
A set of vectors in which none of the vectors can be written as a linear combination of the others.
What is a linearly dependent set?
A set of vectors in which one or more of the vectors can be written as a linear combination of the others.
What is the null space (kernel) of a matrix?
The set of all solutions to the homogeneous system of linear equations Ax = 0, where A is the matrix.
What is the image (range) of a matrix?
The set of all possible outputs of the linear transformation defined by the matrix.