Consider the matrix A = [1 4 3 −2] . (a) Let V be any vector in R2. Explain why V, AV, and A2V must be linearly depen- d
Posted: Thu May 12, 2022 11:06 am
Consider the matrix
A =
[1 4
3 −2]
.
(a) Let V be any vector in R2. Explain why V, AV, and A2V must be
linearly depen-
dent.
(b) Let V = E1 = [1 0]T. Find a non-trivial linear combination of
V, AV, and A2V
that gives the zero vector.
(c) Now let V = E2 = [0 1]T. Find a non-trivial linear combination
of V, AV, and
A2V that gives the zero vector.
(d) Now pick another (random) vector V ∈ R2, and calculate a
non-trivial linear
combination of V, AV, and A2V that gives the 0-vector. What do your
answers
to (b), (c) and (d) have in common?
A =
[1 4
3 −2]
.
(a) Let V be any vector in R2. Explain why V, AV, and A2V must be
linearly depen-
dent.
(b) Let V = E1 = [1 0]T. Find a non-trivial linear combination of
V, AV, and A2V
that gives the zero vector.
(c) Now let V = E2 = [0 1]T. Find a non-trivial linear combination
of V, AV, and
A2V that gives the zero vector.
(d) Now pick another (random) vector V ∈ R2, and calculate a
non-trivial linear
combination of V, AV, and A2V that gives the 0-vector. What do your
answers
to (b), (c) and (d) have in common?