Linear Algebra for Computer Scientists II

Two vectors in Rn are orthogonal if their dot product is zero. This means the angle between them is 90 degrees.

Two subspaces are orthogonal if every vector in one subspace is orthogonal to every vector in the other subspace.

A set of vectors where each vector is orthogonal to all the other vectors in the set.

A set of vectors where each vector is orthogonal to all the other vectors in the set, and each vector has unit length (length 1).

A vector space can be decomposed into a direct sum of orthogonal subspaces.

The process of finding the closest point in a subspace to a given vector.

Finding the best fit solution to an inconsistent system of linear equations, minimizing the error.

A process to create an orthonormal basis from any linearly independent set of vectors.

The projection of a vector b onto a subspace S is the vector in S that minimizes the distance between b and S.

The projection of a vector b onto a subspace S is obtained by finding the vector p in S such that the error vector e = b - p is orthogonal to S.

The projection of a vector onto a subspace is a linear transformation, meaning it preserves vector addition and scalar multiplication.

The projection of a vector onto a subspace spanned by a set of linearly independent vectors can be found by solving a system of linear equations.

The normal equations AT Ax̂ = AT b describe the condition that the error vector is orthogonal to the subspace.

The matrix AT A is invertible if and only if the columns of A are linearly independent.

The projection of a vector onto a subspace can be expressed in terms of the inverse of the matrix AT A.

The projection matrix for projecting any vector onto the column space of A.

It is symmetric, which means it is equal to its transpose.

The projection matrix for projecting any vector onto the row space of A.

The pseudoinverse of A is the transpose of the pseudoinverse of A transpose.

It shows that for any vector in the column space of A, there is exactly one vector in the row space of A that maps to it.

A set of points in Rn defined by a system of linear inequalities with rational coefficients.

A subset of variables in Rn.

The set of points in Rs that can be obtained as projections of points in P.

The projected length of a vector onto another vector or subspace. This represents the component of the vector that lies in the direction of the other vector or subspace.

The length of a vector. In the context of the Gram-Schmidt Process, the vectors are normalized by dividing them by their norms, making them have unit length.

A set of vectors that are all mutually perpendicular to each other, and each vector has a unit norm (length 1).

The process of removing the component of a vector that lies in the direction of another vector or subspace. This leaves only the component that is perpendicular to the other vector or subspace.

A vector that has a length of 1.

A set of vectors that are linearly independent, meaning that no vector can be expressed as a linear combination of the other vectors. In the context of the Gram-Schmidt process, the original vectors are linearly independent, ensuring that the resulting orthonormal basis is valid.

The inverse of a nonsingular square matrix A can be calculated using its cofactors. It's represented as the transpose of the matrix of cofactors, divided by the determinant of A.

The relationship between a nonsingular matrix A, its cofactor matrix C, and its determinant is that the product of A and the transpose of C equals the determinant of A multiplied by the identity matrix.

Cramer's Rule provides a formula to solve a system of linear equations Ax = b where A is nonsingular.

In Cramer's Rule, the solution for each variable (x_j) is calculated by dividing the determinant of a modified matrix (B_j) by the determinant of the original matrix (A).

The modified matrix (B_j) in Cramer's Rule is obtained by replacing the j-th column of A with the vector of constant terms (b).

Cramer's Rule is a method to solve a system of linear equations using determinants. It involves calculating determinants of several matrices to find the values of each variable.

Cramer's Rule is a powerful tool for solving systems of linear equations, particularly when the matrix is invertible. It provides a direct approach using determinants to find the solution for each variable.

While Cramer's Rule provides a formula for solving systems of linear equations, calculating multiple determinants can be computationally expensive. This makes it less efficient for larger systems.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.

The imaginary unit 'i' is a number that satisfies the equation i² = -1. It is used to represent the square root of -1, which is not a real number.

An eigenvalue of a square matrix A is a scalar λ such that there exists a non-zero vector v, called an eigenvector, that satisfies the equation Av = λv. This means that when A multiplies v, it simply scales v by a factor of λ.

An eigenvector of a square matrix A corresponding to an eigenvalue λ is a non-zero vector v that satisfies the equation Av = λv. It represents a direction in which the linear transformation defined by A acts by simply scaling the vector.

The characteristic equation of a square matrix A is the equation det(A - λI) = 0, where λ is an unknown scalar and I is the identity matrix. The solutions to this equation are the eigenvalues of A.

The determinant of a matrix is a scalar value that represents the volume scaling factor of the linear transformation represented by the matrix. It is used to check if a matrix is invertible.

To find eigenvalues and eigenvectors of a matrix A, we first find the characteristic equation det(A - λI) = 0. Solve this equation for λ to get the eigenvalues. For each eigenvalue λ, solve the system of equations (A - λI)v = 0 to find the corresponding eigenvector v.

Eigenvalues and eigenvectors are fundamental concepts in linear algebra with applications in a wide range of fields, including physics, engineering, computer science, and economics.