Let v1 be an eigenvector of an n×n matrix A corresponding to λ1,
and let v2,...
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.
(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 .
Let A∈Rn× n be a non-symmetric matrix.
Prove that |λ1| is real, provided that
|λ1|>|λ2|≥|λ3|≥...≥|λn|
where λi , i= 1,...,n are the eigenvalues of A, while
others can be real or not real.
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 W be a subspace of R^n and suppose that v1,v2,w1,w2,w3 are
vectors in W. Suppose that v1; v2 are linearly independent and that
w1;w2;w3 span W.
(a) If dimW = 3 prove that there is a vector in W that is not
equal to a linear combination of v1 and v2.
(b) If w3 is a linear combination of w1 and w2 prove that w1 and
w2 span W.
(c) If w3 is a linear combination of w1 and...
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) =...
Let G a graph of order 8 with V (G) = {v1, v2, . . . , v8} such
that deg vi = i for 1 ≤ i ≤ 7. What is deg v8? Justify your
answer.
Please show all steps thank you
Let T : Rn →Rm be a linear transformation.
(a) If {v1,v2,...,vk} is a linearly dependent subset of Rn,
prove that {T(v1),T(v2),...,T(vk)} is a linearly dependent subset
of Rm.
(b) Suppose the kernel of T is {0}. (Recall that the kernel of a
linear transformation T : Rn → Rm is the set of all x ∈ Rn such
that T(x) = 0). If {w1,w2,...,wp} is a linearly independent subset
of Rn, then show that {T(w1),T(w2),...,T(wp)} is a linearly
independent...