Linear Algebra TFAE and definitions
If A has linearly independent cols/rows, what is the determinant
!=0
If A has linearly dependent cols/rows, what is the determinant?
0
definition of symmetric matrix
A = transpose(A). Also has all real eigvals
Definition of orthogonal matrix
A'A = AA' = I
If det(A) != 0, what does that mean about Ax=0?
Ax=0 has only trivial solutions
T/F If Ax=b has no soln, then Ax=0 has no soln
False; trivial
Consistent system
Has at least 1 solution
If det(A) != 0 what does that mean about its invertibility
Invertible, A^-1 exists, nonsingular
If Dx=0 has only the trivial solution, what is true about its invertibility
Invertible; 0 is not an eigenvalue
Conditions for something to be a basis
Linear independence Same span as n (for R^n).
Conditions for subspace
Must be closed under vector addition and scalar multiplication (and contain 0 vector, which is included under scalar mult)
Definition of homogeneous system
No constant terms or external forcing terms; no terms that don't depend on unknown functions Homog: x'=Ax Non-homog: x'=Ax+b(t)
What's the stability if the eigenvalues are different signs and are real?
Saddle, unstable
Theorem 7.7.1 (Gram-Schmidt)
Suppose B = [w1,w2,...,wn] is an orthonormal basis for R^n. If u is any vector in R^n, then u=(u*w1)w1 + (u*w2)w2+...+(u*wn)wn Used in Gram-Schmidt to generate an orthogonal basis B' from any given basis B for R^n, then an orthonormal basis B'' = [w1, w2,...,wn] by normalizing B'.
What does G-S do?
Takes already LINEARLY INDEPENDENT set and makes it orthonormal
T/F If Ax=b has unique solution, then Ax=0 has unique soln
True (homogeneous eqn)
Undetermined systems = lin independent or dependent?
Undetermined = dependent
Definition of a linear ODE
Unknown functions and derivatives only appear to the first power, only appear linearly (no sin(x), and the coefficients have no variable dependence except on t Ex: x'1 = -x1+x2 x'2 = 3x1+2x2 Nonlinear: x'2 = sin(x1)+x2^3
switching rows/cols does what to the sign on the determinant
changes +/-
If det(A) != 0, what does that mean about the columns
cols(A) are linearly independent
Is a short fat matrix linearly independent or dependent?
dependent, underdetermined
process to determine stability:
given diff eq, make phase portrait to determine stability near critical point. Does dx/dt converge towards or away from the point?
What's the stability if one eigval = 0 and one is nonzero?
inconclusive, visually investigate
multiplying a row/col by scalar does what to the determinant
multiplies it by the same scalar
What's the stability if the eigenvalues are purely imaginary, +/-iB and a=0?
neutrally stable, center
What's the stability if the eigenvalues are both negative and real?
node, stable
What's the stability if the eigenvalues are both positive and real
node, unstable
How to find rank of a matrix?
number of linearly independent rows/cols. Also number of pivots after rref()
Tall, skinny matrix is
overdetermined
If det(A) != 0 what does that mean about the rank
rank(A)=nrows=mcols
how to find column space
read columns off corresponding to pivot points on rref()
How to find row space
read the rows with pivots (leading "1s" that occur at a later column than the previous row)
If det(A) != 0 what does that mean about rref(A)
rref(A)=In
inverting a matrix does what to the eigenvalues and eigenvectors
same eigenvectors, inverted eigenvalues
What's the stability if the eigenvalues are complex a+/-iB and a<0?
spiral point, asymptotically stable
determinant can only be found of
square matrices
Definition of autonomous ODE
system doesn't explicitly depend on t x'=f(x)
What does it mean for free variables if rank(A)<n
they exist; infinitely many solutions (chemistry problem with unlimited scaling of inputs and outputs)
What's the stability if the eigenvalues are complex a+/-iB and a>0?
unstable spiral point
