Matrix Algebra
This process, named after a famous mathematician can be used to solve arbitrarily large systems of equations using matrices.
What is Gaussian Elimination? (Or Row Reduction)
A vector space that is a subset of some larger vector space
What is a subspace?
Linear transformation T is said to be this if there is a transformation S such that S(T(x))=x
What is invertible?
What lambda in the following equation represents: Ax = λx
What is an eigenvalue?
A process by which two vectors are multiplied together, and the result is a scalar
What is the dot product?
A diagonal matrix where all elements on the diagonal are ones
What is the identity matrix?
This is obtained by taking the sum of scalar multiples of a set of vectors
What is a linear combination?
The two terms in the name of this theorem sum to the number of columns in a given matrix
What is the Rank-Nullity Theorem?
The collection of eigenvectors associated with each eigenvalue generate this
What is an eigenspace?
Two vectors are this if their inner product is zero
What is orthogonal?
The sum of the elements on the main diagonal of a square matrix
What is the trace?
A linearly independent spanning set
What is a basis?
The linear subspace of the preimage that maps to the zero vector. Alternately, the inverse image of 0.
What is the kernel?
Given by this expression: det(A − λI), the roots of which are exactly the eigenvalues of a matrix A.
What is the characteristic polynomial?
A term used to describe a set of orthogonal unit vectors
What is orthonormal?
A matrix that is equal to its own transpose
What is a symmetric matrix?
These two operations are a requirement for all vector spaces
What are vector addition and scalar multiplication?
A type of transformation where the columns of the transformation matrix span the image
What is an "onto" transformation?
This term refers to how many times an eigenvalue appears as a root of the characteristic polynomial
What is multiplicity?
The square root of the inner product of a vector with itself
What is the norm?
A matrix that does not have an inverse
What is a singular matrix?
This property of a vector space is equivalent to the size of its basis
What is dimension?
A type of transformation where the nullity is zero
What is a "one-to-one" transformation?
A process that can be done to a matrix if and only if it's eigenvectors form a basis of n-dimensional realspace
What is diagonalization?
This is a way to turn a finite set of linearly independent vectors into an orthogonal set that spans the same space.
What is the Gram-Schmidt Process?