Linear Algebra Flashcards
What is a vector?
A vector is a quantity that has both magnitude and direction.
True or False: A scalar is a single number that can represent a magnitude.
True
Fill in the blank: The sum of two vectors is called a _______.
resultant vector
What is the geometric representation of a vector in two dimensions?
An arrow from the origin to a point in the plane.
What is the dot product of two vectors?
The sum of the products of their corresponding components.
What is a matrix?
A rectangular array of numbers arranged in rows and columns.
True or False: Matrices can only represent linear transformations.
False
What is the identity matrix?
A square matrix with ones on the diagonal and zeros elsewhere.
What does it mean for a matrix to be invertible?
There exists another matrix such that their product is the identity matrix.
Fill in the blank: The determinant of a matrix provides information about its _______.
invertibility
What is a linear transformation?
A mapping between two vector spaces that preserves vector addition and scalar multiplication.
Define eigenvalue.
A scalar that indicates how much a corresponding eigenvector is stretched or shrunk during a linear transformation.
What is the purpose of row reduction?
To simplify a matrix to its row echelon form or reduced row echelon form.
True or False: A system of linear equations can have no solution, exactly one solution, or infinitely many solutions.
True
What is the rank of a matrix?
The dimension of the vector space generated by its rows or columns.
Fill in the blank: A homogeneous system of linear equations has the form Ax = _______.
0
What is the purpose of the Gram-Schmidt process?
To orthogonalize a set of vectors in an inner product space.
What is a basis in a vector space?
A set of vectors that are linearly independent and span the vector space.
What does it mean if vectors are linearly independent?
No vector in the set can be written as a linear combination of the others.
Fill in the blank: The span of a set of vectors is the _______ of all possible linear combinations of those vectors.
set
What is the relationship between the null space and the rank of a matrix?
The rank-nullity theorem states that the sum of the rank and the nullity of a matrix equals the number of its columns.
True or False: The column space of a matrix is the set of all possible linear combinations of its columns.
True
What is the significance of the characteristic polynomial?
It is used to find the eigenvalues of a matrix.
Define orthogonal vectors.
Vectors that are perpendicular to each other, meaning their dot product is zero.
What is a homogeneous linear equation?
An equation of the form Ax = 0, where A is a matrix and x is a vector.
Fill in the blank: The solution set of a linear system can be described by its _______.
general solution
