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 the largest set of vectors in Rn that has this property.
What is: 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 the smallest set of vectors in Rn that has this property.
What is: 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 properties are required for V to be a subspace of H.
What is:
0) V must be in H
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
Eigenvalues can be these kinds of numbers
What are real?
(For now, we will look at complex eigenvalues and eigenvectors in Chapter 5.3.)
The only vector on the planet that cannot be an eigenvector.
What is the zero vector of any dimension?
The standard basis for Pn = {1, t, t2, ..., tn} 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) The matrix is square (n-by-n)
(2) The matrix is invertible
(3) The columns span Rn
(4) The columns are linearly independent
(5) The determinant is non-zero
(5) The matrix has all non-zero eigenvalues
(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
This type of basis has vectors that are mutually orthogonal and of unit length - like e1, e2, ... en in Rn and
veci , vec j, and vec k
in R3.
What is orthonormal?
The matrix
A - lambdaI
is this kind of matrix
What is singular or non-invertible?
B = {1 + t + t2, -1 + 2t2, 3 - t}
These are the basis vectors
b1, b2,and b3
such that: PB = [b1 b2 b3] and x = PB[x]B
{((1),(1),(1)), ((-1),(0),(2)),((3),(-1),(0))}
Let B={b1, b2, ... bn} and C={c1, c2, ... cn} be bases for Rn. Under what conditions will [x]B = [x]C ?
1) What is x = 0 (the zero vector)?
2) The two bases are the same, that is, B = C
3) The change-of-coordinate matrices
P_(ClarrB) and P_(BlarrC)
have an eigenvalue of 1
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))}
The set of Rn vectors S = {v1, v2, v3, ..., vp} is linearly independent and let A = [v1 v2 ... vp ].
Therefore, we can conclude these facts about:
- S and its vectors
- the matrix A
- the linear transformation T(x) = Ax
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
T:
(1) The mapping x --> Ax is a one-to-one mapping from Rn to Rp
These are eigenvectors of the identity matrix In (i.e.the n-by-n identity matrix)
What is any non-zero vector in Rn?
However, to form a basis for the eigenspace, we would need to choose n linearly independent Rn vectors
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 equation relates the 5 numbers that are generally accepted to be the 5 most important numbers in all of mathematics:
e^(ipi) + 1 = 0
What is Euler's Identity?
0: The additive identity
1: The multiplicative identity
pi: The ratio of the circumference of a circle to its diameter
e: Euler's number, the base of the natural logarithm
bb"i: " sqrt(-1)