Question

Given Eigenvalue 3, -2. Respective Eigenvector V1 = [1 1], V2= [1 -1]. Find the matrix A

Answer #1

v is an eigenvector with eigenvalue 5 for the invertible matrix
A. Is v an eigenvector for A^-2? Show why/why not.

Find two non-parallel vectors v1 and v2 in R2 such that
||v1||2=||v2||2=1 and whose angle with [3,4] is 42 degrees. *Note:
[3,4] is a 2 by 1 matrix with 3 on the top and 4 on the bottom.

Verify that u=[1,13]T is an eigenvector of the matrix
[[ -8,1],[-13,6]]. Find the corresponding eigenvalue lambda.

Let A be an symmetric matrix. Assume that A has two different
eigenvalues ?1 ?= ?2. Let v1 be a ?1-eigenvector, and v2 be and
?2-eigenvector. Show that v1 ? v2. (Hint: v1T Av2 = v2T Av1.)

Let H=Span{v1,v2} and
K=Span{v3,v4}, where
v1,v2,v3,v4 are given
below.
v1 = [3 2 5], v2 =[4 2 6], v3
=[5 -1 1], v4 =[0 -21 -9]
Then H and K are subspaces of R3 . In fact, H and K
are planes in R3 through the origin, and they intersect
in a line through 0. Find a nonzero vector w that
generates that line.
w = { _______ }

Suppose A is a 2x2 matrix with vectors v1=(-12, 10) v2=(-15,13).
Find an invertible matrix P and a diagonal matrix D so that
A=PDP-1. Use your answer to find an expression for
A7 in terms of P, a power of D, and P-1 in
that order.

5. Let v1 = (1/3,−2/3,2/3), v2 = (2/3,−1/3,−2/3) and v3 =
(2/3,2/3,1/3).
(a) Verify that v1, v2, v3 is an orthonormal basis of R 3 .
(b) Determine the coordinates of x = (9, 10, 11), v1 − 4v2 and
v3 with respect to v1, v2, v3.

consider the basis S={v1,v2} for R^2,where v1=(-2,1) and
v2=(1,3),and let T:R^2-R^3 be linear transformation such that
T(v1)=(-1,2,0) And T(v2)=(0,-3,5), find T(2,-3)

Let A =
v1
v2
v3
v1
2
8
16
v2
8
0
4
v3
16
4
1
be the ADJACENCY MATRIX for an undirected graph G. Solve the
following:
1) Determine the number of edges of G
2) Determine the total degree of G
3) Determine the degree of each vertex of G
4) Determine the number of different walks of length 2 from
vertex v3 to v1
5) Does G have an Euler circuit? Explain

The matrix A has an eigenvalue λ with an algebraic multiplicity
of 5 and a geometric multiplicity of 2. Does A have a generalised
eigenvector of rank 3 corresponding to λ? What about a generalised
eigenvector of rank 5?

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