Exam 1 Flashcards
tr(A)
sum of main diagonal of square matrix
row equivalent
you can get from A to B using only elementary row operations
(A^-1)^-1 =
A
(A^n)^-1 =
A^-n = (A^-1)^n
(αA)^-1 =
(1/α)A^-1
A^r A^s =
A^(r+s)
(A^r)^s =
A^(rs)
(A^T)^T =
A
(A+B)^T =
A^T+B^T
(αA)^T =
αA^T
(AB)^T =
B^T A^T
If you get a zero row while inverting a matrix,
it is not invertible
A is symmetric if
A^T=A
M_i,j(A) =
det of matrix gotten from A by removing row i and column j.
C_i,j(A) =
(-1)^(i+j) M_i,j(A)
det(αA) =
α^n det(A)
det(A+B) ≠
detA+detB
detAB =
detAdetB
Biggerer Theorem
(a) A is invertible
(b) Ax=0 has only trivial solution
(c) A is row equivalent to I
(d) A is a product of elementary matrices
(e) Ax=b is consistent for any b
(f) Ax=b has only one solution for any b
(g) detA≠0
detA^-1 =
1/detA
adj(A) =
(CofA)^T
Aadj(A) =
(detA)I
Cramer’s Rule
For Ax=b and A is square, if detA≠0 then x_i=det(A_i)/det(A) where A_i=A with column i replaced by b.
A vector is defined by
its end point minus its start point
u+v =
v+u
u+(v+w) =
(u+v)+w
u+0=
u
u-u =
0
α(u+v)=
αu+αv
(α+β)u=
αu+βu
α(βu)=
(αβ)u
u·v =
v·u
u·(v+w) =
u·v + u·w
α(u·v) =
(αu)·v = u·(αv)
Norm (length) of v =
||v|| = √(v·v)
||αv|| =
|α| ||v||
normalization of v =
(1/||v||)v
distance d(u,v)=
||u-v|| = ||v-u||
cosθ =
u·v if u and v are unit vectors
Cauchy-Schwartz Inequality
|u·v| ≤ ||u|| ||v||
u and v are orthogonal if
u·v = 0
(proj_u)v =
(u·v)u if u is a unit vector
(proj_u^⊥)v =
v - (proj_u)v
Line through point P in the direction of v
r(t) = P + tv
Plane through point P in the direction of u and v
x(s,t) = P + su + tv
