(a) If V1,V2⊂V show that (V2^⊥)⊂(V1^⊥) implies V1⊂V2
(b) If V1,V2⊂V , show that (V1+V2)^⊥=(V1^⊥)∩(V2^⊥) where we
write V1+V2 to be the subspace of V spanned by V1 and V2 .
If G = (V, E) is a graph and x ∈ V , let G \ x be the graph
whose vertex set is V \ {x} and whose edges are those edges of G
that don’t contain x.
Show that every connected finite graph G = (V, E) with at least
two vertices has at least two vertices x1, x2 ∈ V such that G \ xi
is connected.
v1=[0,1,4] v2=[-4,-5,7] v3=[14,10,8] b=[16,18,19].
Let v1,v2, and v3 be three nonzero vectors in R3. Suppose v2 is
not a scalar multiple of either v1 or v3 and v3 is not a scalar
multiple of either v1 or v2. Does it follow that every vector in R3
is in span{v1,v2,v3}?
Let v1 be an eigenvector of an n×n matrix A corresponding to λ1,
and let v2, v3 be two linearly independent eigenvectors of A
corresponding to λ2, where λ1 is not equal to λ2. Show that v1, v2,
v3 are linearly independent.
let l be the linear transformation from a vector space V where
ker(L)=0 if { v1,v2,v3} are linearly independent vectors on V prove
{Lv1,Lv2,Lv3} are linearly independent vectors in V
Let v1 = [-0.5 , v2 = [0.5 , and
v3 = [-0.5
-0.5 -0.5 0.5
0.5 0.5 0.5
-0.5] 0.5] 0.5]
Find a vector v4 in R4 such that the
vectors v1, v2, v3, and
v4 are orthonormal.
16. Which of the following statements is false?
(a) Let S = {v1, v2, . . . , vm} be a subset of a vector space V
with dim(V) = n. If m > n, then S is linearly dependent.
(b) If A is an m × n matrix, then dim Nul A = n.
(c) If B is a basis for some finite-dimensional vector space W,
then the change of coordinates matrix PB is always invertible.
(d) dim(R17) =...
If {v1, v2, v3, v4} is a linearly-independent subset of a
vector space V over the field Q, is the
set {3v1 + 2v2 + v3 + v4, 2v1 + 5v2, 3v3 + 2v4, 3v1 + 4v2 +
2v3 + 3v4} linearly independent as well?
Let G = (V, E) be a directed graph, with source s ∈ V, sink t ∈
V, and nonnegative edge capacities {ce}. Give a polynomial-time
algorithm to decide whether G has a unique minimum s-t cut (i.e.,
an s-t of capacity strictly less than that of all other s-t
cuts).