Ax=b
These are the two primary methods for solving a system of equations Ax=b
What are:
1) Row reducing the augmented matrix [A | b]
2) If A is invertible, x= A-1b
For an mxn matrix A, vectors in Col A and Nul A are elements of these vector spaces respectively
What is Rm and Rn?
For the system
A/(mxn) bbx/(nx1) = bb b/(mx1)
Col A: Column space vectors are the set of all possible "b-vectors" and thus belong to Rm
Nul A: Null space vectors are solutions, or "x-vectors", to the homogeneous system Ax=0 and thus belong to Rn.
The linear transformation T(x)=Ax, where A is an mxn matrix, is a mapping from this vector space to this vector space.
What is from Rn to Rm?
Rationale:
A/(mxn) bbx/(nx1) = bb b/(mx1) implies T: R^n rarr R^m
These are the primary steps in solving the IVP
x'(t) = Ax(t) for a given initial condition
x=((x_1(0)), (x_2(0)))
What is:
1) Determine the eigenvalues and corresponding eigenvectors of A; CYFA
2) Construct the general solution:
bbx(t) = c_1bbv_1e^(lambda_1t) + ... c_nbbv_e^(lambda_nt)
3) Apply the initial condition x(0) to solve for c1, ... cn
4) CYFA - verify that your x(0) agrees with the given initial condition
The process of "normalizing" a vector by its magnitude
bbx/norm(bbx)
produces a vector with this name.
What is a unit vector?
If there is a solution to the equation Ax=b for all b in Rm, then we can say this about the mxn matrix A.
What is:
1) A has a pivot in every row
2) The columns of A spans Rm
NOTE: While it is true that the system will be consistent for all b, this is not a property of A, being consistent or inconsistent is a property of a SYSTEM of equations not a matrix. Similarly, we say that the COLUMNS of A span Rm, not A itself.
A plane containing the origin in any vector space Rn is isomorphic to this vector space.
What is R2?
A plane through the origin is a 2-dimensional subspace of ANY vector space it resides in, and is thus isomorphic to R2.
This property of an mxn matrix A guarantees that the mapping defined by the linear transformation T(x)=Ax is one-to-one.
What is that the columns of A are linearly independent?
Rationale: Chapter 1.9 Theorem 12b
Alternate answers:
1) What is that the equation Ax=0 has only the trivial solution (Chapter 1.9 Theorem 11)
2) What is that an echelon form a A has a pivot in every COLUMN ==> no free variables
True or False: The eigenvector that x(t) will try to align with as
t -> oo
is always in the direction of maximum attraction or repulsion relative to the origin?
What is FALSE?
This is TRUE for the REPELLER and SADDLE POINT cases. However, the counterexample is the ATTRACTOR case where the direction of maximum attraction to the origin is parallel to the eigenvector that the solution does NOT align with because its exponential decays the fastest. You can see this in the trajectory, or algebraically if we enumerate the eigenvalues such that
lambda_1<lambda_2<0
we can see that initially the first term in the velocity function x'(t) has a greater negative velocity than the first term (the "speed" in the direction of -v1 is greater than the speed in the direction of -v2)
x'(t)=c_1bbv_1lambda_1e^(lamda_1t)+c_2bbv_2lambda_2e^(lamda_2t)
It is true that eventually the speed in the -v2 direction will overtake the speed in the -v1 direction from that time forward.
Vectors u and v are defined to be orthogonal if they have this property.
What is that their inner (dot) product
bbucdotbbv=0?
NOTE: It is true that orthogonal vectors in R2 and R3 are perpendicular (i.e. have an angle of
Pi/2
between them). However, this is NOT the definition of orthogonality and perpendicularity does not readily extend beyond R3.
For a square matrix A, this vector is assured to be in both the null space and column space of A. That is, it can be a "x-vector" solution for Ax=0 or serve as a "b- vector" in a consistent system Ax=b
What is the zero vector?
Subtle but important point: The zero vector will ALWAYS be in both the null space and column space of a matrix. However, if the matrix is not square, that is if
mnen
then the two zero vectors will NOT be the same vector because they would be in different vector spaces. (Rn and Rm respectively for null space and column space)
The vector space of polynomials Pn is isomorphic to Rk where k is this value.
What is k = n+1?
Rationale: An nth degree polynomial has n+1 coefficients.
This property of an mxn matrix A guarantees that the mapping defined by the linear transformation T(x)=Ax is onto Rm
What is that the columns of A span Rm?
Rationale: Chapter 1.9 Theorem 12a
Alternate answer: An echelon form A has a pivot in every ROW.
Of the three cases we have studied, Attractor, Repeller, and Saddle Point, this is the one case where the solution does NOT approach the dominant vector asymptotically.
What is the REPELLER case?
BOTH terms become unbounded so while the solution follows the general direction of the vector v2 (the one with the largest eigenvalue), the distance between the solution and the guideline parallel to v2 actually grows with time.
If the R3 vectors b1, b2, and b3 are orthogonal, then the set
ccB={bb b_1,bb b_2,bb b_3}
is called this.
What is an orthogonal basis for R3?
Alternate Answer:
1) What is an orthogonal set?
Note that orthogonal vectors are linearly independent, and since there are exactly 3 of them, we can make the stronger statement that the set forms a basis for R3. However the converse is not true - a set of basis vectors for R3 vector is not necessarily an orthogonal set.
This property of an mxn matrix A guarantees that the system Ax=b will never be inconsistent
What is that A spans Rm?
Alternate answers:
1) What is that an echelon form of A has a pivot in every row?
2) What is A is invertible? (Only applicable when m = n)
True or False: For any n<m, Rn is a subspace of Rm
What is FALSE?
Vectors in Rn for any
n ne m
do even not belong to Rm so they cannot form a subspace of Rm.
If for the linear transformation T(x)=Ax there exists an inverse transformation S such that S(T(x)) = x and T(S(x))=x, then we can say this about the matrix A.
What is A is invertible?
Additional answers:
What is that A is square (nxn) and the columns of A form a basis for Rn or any of the other gazillion equivalent conditions of the IMT?
If a solution trajectory approaches a guideline parallel to v2 asymptotically from above, the trajectory will have this concavity.
Similarly, if the trajectory approaches the guideline from below, it will have this concavity.
What are concave UP and concave DOWN respectively?
If the R3 vectors b1, b2, and b3 are orthogonal and ||b1||=||b2||=||b3||=1, then the set
ccB={bb b_1,bb b_2,bb b_3}
is called this.
What is an orthonormal basis for R3?
For example, the R3 vectors e1, e2, and e3 form an orthonormal basis for R3. Or using Calc III notation, these would be i, j, and k respectively.
For a non-square mxn matrix A, if the system Ax=b has non-trivial solutions then we can say this about m and n.
What is n > m; or if not then the columns must form a dependent set?
Rationale: If the system has non-trivial solutions there must be at least one free variable. The only way to guarantee this for all A is to have more columns than rows. And if that is not the case, then the columns must be linearly dependent.
These four properties are required for a set of vectors S to be a subspace of V.
What is that S:
Is a subset of V (a given in the definition)
Definition Conditions:
a) Contains the zero vector
b) Is closed under vector addition
c) Is closed under scalar multiplication
A linear transformation that is a change-of-coordinates between two bases
cc "B" and cc"C"
within a vector space such as Rn, will always have these two properties and for this reason.
What is that the mapping is 1-to-1 and ONTO because the change-of-coordinates matrices
P_(cc"B"larrcc"C") and P_(cc"C"larrcc"B")
will ALWAYS be invertible since they are comprised of basis vectors and thus IMT condition (m) applies.
For a 2-by-2 system x'(t) = Ax(t), if one, but not both, of the eigenvalues is zero, we can say this about:
1) The reference point for characterizing the behavior as an Attractor, Repeller, or Saddle Point
2) The eigenvector that the solution will align with
What is:
1) The origin is no longer the reference point for being an Attractor, Repeller, or Saddle Point. The reference point will now be the "tip" of the eigenvector with the zero eigenvalue.
2) The dominant eigenvector will be the one with the non-zero eigenvalue.
Given a vectors u and x,
bbhaty=((bbycdotbbu)/(bbucdotbbu))bbu =(bbycdotbbu/||bbu||)bbu/||bbu||
is called this.
Note
bbu/||bbu||
is a unit vector in the direction of u.
What is the orthogonal projection of x onto u?
Or, as the book states, the orthogonal projection of x onto L where L is the subspace (line through the origin) containing u.