MATH 2270 PRACTICE PROBLEMS FOR MIDTERM 3
1. Practice Exam
1) Show that P2is isomorphic to F3.
2) Suppose a 5 ×7 matrix, A, has three pivot columns. What is the
dimension of the null space of A? Is F3the column space of A? Justify
your answers.
3) Let A=1 3
1−1. Find the eigenvalues of Aand a non-zero
vector from each eigenspace.
4) Find a diagonal matrix that is similar to
1−1 0
3−1 0
005
.
5) Suppose Ais a square matrix with real entries. Show that if λis
an eigenvalue of A, then so is ¯
λ.
6) Consider the equation ˙
x=Axwhere xis a vector valued function
of tin R2and A=4 1
3 2 . It can be show that the general solution
to the equation is given by x(t) = cetλv+detµuwhere λand µare the
eigenvalues of Awith corresponding eigenvectors vand urespectively,
and cand dare arbitrary constants. Find the general solution of the
equation.
7) Compute the distance between
1
−3
2
5
and
2
2
−1
7
.
8) Let u=
1
3
−2
,v=
5
1
4
. Let y=
1
3
5
. Show that uand
vare orthogonal. Write yas the sum of a vector in Span{u,v}=W
and a vector in W⊥.
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