Eigen Schmeigen
This, and only this, type of matrix has an eigenvalue of 0
What is singular or non-invertible?
A basis for Rn is a largest set of vectors in Rn that has this property.
What is that the set is linearly independent?
Basis
What is the best pun for the word?
These are the subspaces we typically associate with a matrix A
What are:
(1) the column space; Col A
(2) the null space; Nul A
(3) the row space; Row A
Bonus: The eigenspace associated with eigenvalues of A
This common problem solving technique is NOT appropriate for finding eigenvalues
What is row reduction?
Eigenvalues for these matrices appear along the diagonal
What is a triangular matrix?
(Alternate answer: What is a matrix in echelon form or in REF?)
A basis for Rn is a smallest set of vectors in Rn that has this property.
What is that the set spans Rn?
What is the best pun?
These three properties are required for V to be a subspace.
What is:
1) V must contain the zero vector
2) V must be closed under vector addition
3) V must be closed under multiplication by a scalar
If A is a diagonalizable matrix, then the inverse of A has this property.
What is diagonalizable?
The only vector on the planet that cannot be an eigenvector.
What is the zero vector of any dimension?
This is the value of the algebraic multiplicity of an eigenvalue of A if A is a nxn matrix and there are n distinct eigenvalues.
What is 1?
Span
What is the best pun?
The maximum and minimum dimension of the null space for a non-zero 4-by-6 matrix A.
What is
2 <= dim "Nul" (A) <= 5?
Max of 4 pivots ==> min dim Nul A = 6 - 4 = 2
Min of 1 pivot ==> max dim Nul A = 6 - 1 = 5
The dimension of the eigenspace corresponding to an eigenvalue of a matrix is bounded above by this.
What is the algebraic multiplicity of the eigenvalue?
The matrix
A - lambdaI
is this kind of matrix
What is singular or non-invertible?
If Rn has a basis consisting of eigenvectors of a matrix A, then A is said to be this.
What is diagonalizable?
Determinant
What is the best pun for this?
A basis for the column space of A
A = [[1,4,5,6],[2,4,6,8],[3,4,7,10], [4,4,8,12]] ~ [[1,0,1,2],[0,1,1,1],[0,0,0,0], [0,0,0,0]]
What is:
{((1),(2),(3),(4)), ((4),(4),(4),(4))}
Let S = {v1, v2, v3, ..., vp} be a linearly independent set of vectors in Rnand let A = [v1 v2 ... vp ].
We can conclude these three facts about S and its vectors, and these three facts about A.
What is:
S:
(1) p<=n
(2) S does not contain the zero vector
(3) None of the vectors can be constructed from a linear combination of any of the other vectors
A:
(1) If n = p then then A is invertible
(2) Any echelon form of A will have a pivot in every column
(3) Ax=0 has only the trivial solution
These are eigenvectors of the identity matrix In (i.e.the n-by-n identity matrix)
What is any non-zero vector in Rn?
Let V be a subspace of Rn with dim V = p. A set of basis vectors for V will have these 4 key properties.
What is:
(1) they will all be elements of Rn
(2) there will be exactly p vectors in the set
(3) the set will be linearly independent
(4) the set will span V
Eigenspace
What is the best pun for this?
A basis for the row space of A
A = [[1,4,5,6],[2,4,6,8],[3,4,7,10], [4,4,8,12]] ~ [[1,0,1,2],[0,1,1,1],[0,0,0,0], [0,0,0,0]]
(1, 0, 1, 2), (0, 1, 1, 1)
This is another name for the dimension of the column space of a matrix.
What the rank of the matrix?