Linear Algebra: Fundamentals

What is the result of the dot product of two orthogonal vectors?

The dot product of two orthogonal vectors is zero.

Define the transpose of a matrix A.

The transpose of a matrix A, denoted as A^T, is formed by swapping its rows and columns.

What is a unit vector?

A unit vector is a vector with a magnitude (length) of exactly 1.

What does the identity matrix I represent in matrix multiplication?

The identity matrix acts as the multiplicative identity, where AI = IA = A for any square matrix A.

What is an augmented matrix in the context of linear systems?

An augmented matrix is a matrix formed by appending the columns of two matrices, typically the coefficient matrix and the constant vector.

How is the magnitude (norm) of a vector v = [x, y, z] calculated?

The magnitude is calculated as ||v|| = sqrt(x^2 + y^2 + z^2).

For the matrix product AB to be defined, what condition must the dimensions of A and B satisfy?

The number of columns in matrix A must equal the number of rows in matrix B.

What is the formula for the determinant of a 2x2 matrix [[a, b], [c, d]]?

The determinant is calculated as det(A) = ad - bc.

When is a set of vectors considered linearly independent?

A set of vectors is linearly independent if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is the trivial solution where all scalars c are zero.

What are the two requirements for a set of vectors to be a basis for a vector space V?

The vectors must be linearly independent and they must span the entire vector space V.