Linear algebra

Here are some example questions under linear algebra:

Given two matrices A and B, find their sum A+B and their product AB.

Find the determinant of a given matrix A and use it to determine if the matrix is invertible.

Given a set of vectors, determine if they are linearly independent or dependent.

Find the basis and dimension of a vector space.

Find the inverse of a given invertible matrix A.

Find the eigenvalues and eigenvectors of a given matrix A.

Given a linear transformation T, determine if it is one-to-one or onto.

Find the kernel and image of a linear transformation T.

Given a matrix A, use row operations to find its reduced row echelon form and solve a system of linear equations.

Find the orthogonal projection of a vector onto a subspace.

These are just a few examples of the many questions that can be asked in linear algebra. Depending on the level of difficulty and the specific focus of the course or exam, the questions can be more or less complex.

Here's an explanation of each of the questions listed in my previous response:

Given two matrices A and B, find their sum A+B and their product AB.

This question involves basic matrix operations. The sum of two matrices A and B is obtained by adding their corresponding entries. The product of two matrices A and B is obtained by multiplying their entries and summing over appropriate indices.

Find the determinant of a given matrix A and use it to determine if the matrix is invertible.

The determinant of a matrix A is a scalar value that can be computed using various methods. If the determinant is nonzero, the matrix is invertible; otherwise, it is singular.

Given a set of vectors, determine if they are linearly independent or dependent.

A set of vectors is said to be linearly independent if none of the vectors can be written as a linear combination of the others. A set of vectors is linearly dependent if at least one of the vectors can be written as a linear combination of the others.

Find the basis and dimension of a vector space.

A basis for a vector space is a set of linearly independent vectors that spans the space. The dimension of a vector space is the number of vectors in any basis for the space.

Find the inverse of a given invertible matrix A.

The inverse of a matrix A is a matrix A^-1 such that A * A^-1 = I, where I is the identity matrix. The inverse can be computed using various methods, such as Gaussian elimination or the adjugate formula.

Find the eigenvalues and eigenvectors of a given matrix A.

An eigenvalue of a matrix A is a scalar value λ such that there exists a nonzero vector x that satisfies Ax = λx. Such a vector x is called an eigenvector of A corresponding to the eigenvalue λ.

Given a linear transformation T, determine if it is one-to-one or onto.

A linear transformation T is one-to-one if distinct inputs yield distinct outputs; that is, if T(x) = T(y) implies x = y. A linear transformation T is onto if every element of the target space is the image of at least one element of the domain.

Find the kernel and image of a linear transformation T.

The kernel of a linear transformation T is the set of all inputs that are mapped to zero; that is, ker(T) = {x : T(x) = 0}. The image of a linear transformation T is the set of all outputs that are obtained by applying T to elements of the domain; that is, im(T) = {T(x) : x is in the domain of T}.

Given a matrix A, use row operations to find its reduced row echelon form and solve a system of linear equations.

Row operations are elementary operations that can be performed on the rows of a matrix to transform it into a simpler form. The reduced row echelon form of a matrix is a specific form that makes it easy to solve systems of linear equations.

Find the orthogonal projection of a vector onto a subspace.

The orthogonal projection of a vector v onto a subspace S is the closest vector in S to v. It can be computed using the formula proj_S(v) = (v . u)u, where u is any nonzero vector in S and "." denotes the dot product.