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?
Let B={b1, b2, b3} be a basis for R3. The matrix
[b1 b2 b3] performs this function.
What is transforms a B-coordinate vector to standard coordinates? That is:
x = PB[x]B where PB =[b1 b2 b3] is the change-of-coordinates matrix from B to standard coordinates
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?
[b1 b2 ... bn | c1 c2 ... cn] ~ [In | THIS MATRIX]
What is the change-of-coordinates matrix from C to B, denoted by
P_(BlarrC)
?
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 characteristic polynomial of A has this property.
What is factors completely into linear factors?
The only vector on the planet that cannot be an eigenvector.
What is the zero vector of any dimension?
The vector space Pn of polynomials of degree less than or equal to n is isomorphic to this vector space.
What is Rn+1?
Because change-of-coordinate matrices within Rn are made up of basis vectors, they have these key important properties (name as many as possible)
What is:
(1) They are square (n-by-n)
(2) They are invertible
(3) They span Rn
(4) The columns are linearly independent
(5) Zero is not an eigenvalue
(6) ... All of the other properties in the IMT
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?
If V is an n-dimensional vector space, the coordinate mapping from V to Rn has these properties:
What is:
1) It is 1-1 and onto
2) It is linear.
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
The linear transformation T(x) = Ax maps all of R3 to a plane in R3. The matrix A has this property.
What is singular (or non-invertible)?
Alternate Answers:
(1) Has a determinant of 0
(2) Has an eigenvalue of 0
(3) There are two pivot columns and one non-pivot column
(4) dim Col A = Rank A = 2; and dim Nul A = 1
(5) Any other property in the IMT is FALSE
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?