Complex Flashcards
What is the definition of a vector?
A vector is a quantity defined by both magnitude and direction.
True or False: A scalar is a single number that represents magnitude only.
True
Fill in the blank: The solution set of a linear equation can be represented as a _______.
line
What is the purpose of a matrix in linear algebra?
A matrix is used to represent and solve systems of linear equations.
What is the determinant of a 2x2 matrix?
For a matrix [[a, b], [c, d]], the determinant is ad - bc.
What does it mean for vectors to be linearly independent?
Vectors are linearly independent if no vector in the set can be expressed as a linear combination of the others.
What is the rank of a matrix?
The rank of a matrix is the maximum number of linearly independent row or column vectors in the matrix.
Multiple choice: What is an eigenvalue?
A scalar associated with a linear transformation that stretches or compresses a vector.
True or False: The inverse of a matrix exists only if the matrix is square.
True
What is a linear transformation?
A function between vector spaces that preserves the operations of vector addition and scalar multiplication.
Fill in the blank: The _______ of a matrix is the matrix obtained by swapping its rows and columns.
transpose
What is a null space of a matrix?
The null space is the set of all vectors that, when multiplied by the matrix, yield the zero vector.
What is the formula for the dot product of two vectors?
The dot product of vectors A and B is A·B = |A||B|cos(θ), where θ is the angle between them.
True or False: A square matrix with a determinant of zero is invertible.
False
What is the geometric interpretation of eigenvectors?
Eigenvectors represent directions in which a linear transformation acts by stretching or compressing.
Multiple choice: Which of the following is a property of symmetric matrices?
The matrix is equal to its transpose.
What is the purpose of Gaussian elimination?
To solve systems of linear equations or to find the rank of a matrix.
Fill in the blank: An n x n matrix has _______ eigenvalues at most.
n
What is a basis in a vector space?
A set of linearly independent vectors that span the vector space.
True or False: Every vector space has a unique basis.
False
What is the characteristic polynomial of a matrix?
A polynomial that is derived from the determinant of the matrix subtracted by λ times the identity matrix.
Multiple choice: What is the primary use of singular value decomposition?
To factor a matrix into its constituent components for analysis.
What does the term ‘orthogonal’ mean in the context of vectors?
Two vectors are orthogonal if their dot product is zero.
Fill in the blank: The _______ theorem states that every linear equation can be represented in matrix form.
matrix representation
What is the relationship between a matrix and its adjugate?
The adjugate of a matrix is the transpose of its cofactor matrix and is used to find the inverse.
What is the concept of vector space?
A vector space is a collection of vectors that can be added together and multiplied by scalars.
