Final Exam, December 17-23, 2020
Final Exam
Linear Algebra, Dave Bayer, December 17-23, 2020
[1]Find an orthogonal basis for R4that includes the vector (1, 0, 0, 1).
[2]Find a system of equations having as solution set the image of the following map from R3to R4.
w
x
y
z
=
0 2 2
1 0 −1
−1−2−1
0−2−2
r
s
t
[3]Let Vbe the subspace of R4defined by the system of equations
1 1 1 0
1 2 2 1
0 1 1 1
w
x
y
z
=
0
0
0
Find the 4×4matrix Athat projects R4orthogonally onto V.
[4]Find Anwhere Ais the matrix
A=0 3
1 2
[5]Find eAt where Ais the matrix
A=
1 1 1
0 1 0
2 2 2
[6]Solve the differential equation y0=Ay where
A=4−1
4 0 ,y(0) = 0
1
[7]Express the quadratic form
3x2+3y2+2xz −2yz +2z2
as a sum of squares of orthogonal linear forms.
[8]Let f(n)be the determinant of the n×nmatrix in the sequence
[ ] 33 1
0 3
3 1 0
0 3 1
−4 0 3
3 1 0 0
0 3 1 0
−4 0 3 1
0−4 0 3
3 1 0 0 0
0 3 1 0 0
−4 0 3 1 0
0−4 0 3 1
0 0 −4 0 3
For example, f(4) = 57 and f(5) = 135.
Find a recurrence relation for f(n).
Using Jordan canonical form, solve this recurrence relation to find a closed form formula for f(n).