A physicist has plotted
the position of a projectile over time. Based
on Newton’s laws, the projectile should ideally travel in a parabolic path.
Use matrices to find the most likely equation of this parabola.
(You may use a calculator, but show the matrices that are being
manipulated).
A
new species of fish is introduced into the Truckee River.
Initially 2 fish were stocked.
It takes one year for this species to spawn, when each fish averages 3
successful children each year. (So
there are 2 at the beginning, 2
at the end of the first year, 8 at the end of the
second year, 14 at the end of the third year, etc.)
A.
Assuming no fish die, set up a recursion relationship that gives the
number of fish w B.
Find a matrix A such that w C. Find a diagonal matrix D such that A is similar to D.
Let
V be the subspace of differentiable functions
spanned by {e
L: V
---> V
be the linear transformation with
L(f(x)) = f ''(x) - 3f '(x) + 2f(x) A.
Write down the matrix A B.
Find the a basis for the kernel and range of L.
Let W
= Span{(1,1,0,1), (0,1,2,3)}. Find
a basis for the orthogonal complement of W.
Let
and T
be the affine transformation T(x) = Ax +
Let L: V ---> V be a linear transformation. Use the fact that
dim(Ker L) + dim(Range L) = dim(V)
to show that if L
is one to one then L is onto.
Let A
and B be matrices and let v
be an eigenvector of both A and B.
Prove that
Answer True of False and
explain your reasoning. A. Let A be a 3x3 matrix such that the columns of A form an orthonormal set of vectors. Then
B. Let V be the vector space of continuous functions then the expression <f, g> = f(1) + g(1) defines an inner product on V. |