Linear Algebra Flashcards
V is finite dimensional if and only if?
It contains a finite spanning set
What does a subset W of V have to satisfy for it to be a subspace?
0 is in W
w,v in W implies that w+v is in W
a real, w in W implies that aw is in W
When is f:V–>W a linear map?
If for all u,v in V, a complex, f(u+av) = f(u) + af(v)
What is the kernel of f?
{v in V | f(v) = 0}
What is rank(f)?
dim(im(f))
What is nullity(f)?
dim(ker(f))
What is the rank nullity formula for f linear and V, W finite dimensional?
dim(V) = rank(f) + nullity(f)
When is λ an eigenvalue of A?
If there exists a non-zero poly in C^n such that Av= λv
When is v in C^n an eigenvector of A?
If it is non-zero and there exists λ in C such that Av = λv
Definition of an innner product function on V ?
For all u,v,w in V, k in R or C, <u> = <u> + k
For all u,v in V <u> = (real) or = conjugate (complex)
For all v in V, >=0, equality only if v=0</u></u></u>
What is ||v||?
The sqrt of
What is the characteristic polynomail, pa(x)?
det(xI - A)
When is A similiar to B?
If there exists an invertible matrix Q st QBQ^-1 =A
What is the Cayley Hamilton theorem?
Pa(A) =0
What is the adjoint of A?
A* = A conjugate transpose
What does it mean if A is self adjoint?
A=A*
What does it mean if U is unitary?
UU* = U*U = I
What does it mean if A is normal?
AA* = A*A
Whenis the matrix A diagonisable?
If and only if there is a basis for C^n consisting of eigenvectors of A
What is the dimension?
The number of linearly independent vectors in a basis
What is im(f)?
{w in W | there exists v in V st f(v) = w}
What is the minimal polynomial a divisor of?
The characteristic polynomial
What is the minimal polynomial?
The monic non-zero poly of minimal degree st ma(A) = 0
What is the rank and nullify of a matrix in REF?
The nullity is the number of zero rows, the rank is the number of lin ind non-zero rows
When is a set orthonormal?
If = 1 when i=j, 0 if i is not equal to j
What is the Pythagorean theorem if <u> =0</u>
||u+v||^2 = ||u||^2 + ||v||^2
What is the Cauchy Schwartz inequality?
|<u>| =< ||u|| ||v||</u>
What is the triangle inequality?
||u+v|| =< ||u|| + ||v||
When is a complex matrix unitary/a real matrix orthogonal?
Iff it’s columns form an orthonormal basis
A matrix is symmetric iff?
It is diagonalizable
What does it mean if A is symmetric?
A = A transpose
A is unitarily diagonalizable iff?
A is normal
What does it mean for A to be orthogonal
AA^t = A^tA = I
What is the transition matrix from A to the standard basis?
A
What is the transition matrix from the standard basis to A?
A^-1
What is the transition matrix from A to B?
(B^-1)A
u is orthogonal to v if?
<u> =0</u>
What is the geometric multiplicity nλ(A)?
The nullity of λI - A, ie. The number of zero rows in the REF
What is the multiplicity mλ(A)?
The number of times that that λ appears in the characteristic polynomial
