Math 2250 Lab 8
Name/Unid:
1. (25 points) Suppose that A is an n × n matrix and that k is a constant real number.
We defined the set of all vectors x that satisfy the equation
Ax = kx.
(a) Show that the set forms a subspace of Rn
(b) Suppose we define A to be

−4
−16
A=
 −7
−11
1
3
2
1
1
4
2
3

1
4

1
4
and let k = 1. Find a basis that spans the subspace Ax = x. What dimension is
the subspace? Hint, you know from Theorem 1 of 4.2 that Systems of the form
Bx = 0 define subspaces. Use the rules of matrix algebra to transform Ax = x into
Bx = 0, then solve for the solution space and find a spanning set. Double hint: the
identity matrix I is useful in this task. Triple hint: use the MATLAB command
”rref(B)” to aid in the row operations to save time and ensure accuracy.
2. (25 points) Vector spaces of functions: Let P be a set of polynomials:
P = {p1 (x), p2 (x), p3 (x), p4 (x)} = {1, x, 3x2 − 1, 5x3 − 3x}
(a) Determine if the set P is linearly independent or not.
(b) Find a linear combination of P that represents the polynomial y(x) = 1+x+x2 +x3
(c) Does P span the vector space V of all polynomials of third-degree or less? Justify
your answer.
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3. (25 points) Consider the differential equation: mx00 + kx = 0.
q
k
(a) Verify that cos(ωt) and sin(ωt), where ω = m
, are linearly independent solutions
to the DE.
(b) Solve the DE with initial conditions: y(0) = 1 and y 0 (0) = 0.
(c) Show that sin(ωt) and sin(ωt + π) linearly independent solutions of the DE.
(d) Use sin(ωt) and sin(ωt + π) to solve the IVP from part b.
(e) Double check that your solution from parts b and d are equivalent using the anglesum identity.
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4. (25 points) The Bessel differential equation is defined as the following differential operator B for x > 0 and n an integer
n2 1
B(y) = y 00 + y 0 + 1 − 2 y = 0.
x
x
Show that the differential equation is linear.
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