Flashcards on Linear Algebra

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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.

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.

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