Math310: Sample Problems from Chapters 5 and 6
The following topics from Chapters 5 and 6 will be covered on the final exam
•Inner product spaces: finding angles, scalar and vector projections; deciding
whether a set of vectors is an orthonormal/orthogonal set; Gram-Schmidt or-
thogonalization process; projecting a vector onto a subspace.
•Rnwith the standard scalar product: deciding whether two subspaces are orthog-
onal; finding orthogonal complement; least squares problems (projecting a vector
onto a subspace).
•Eigenvalues and eigenvectors: finding eigenvalues and eigenspaces of a matrix; de-
ciding whether a matrix is diagonalizable; finding a representation A=XDX −1
when possible; computing rational powers of a matrix and matrix exponential eA;
solving systems of linear differential equations.
Problem 1. Find the eigenvalues and the corresponding eigenspaces for each of the
following matrices. Decide whether each of the following matrices is diagonalizable or
not. If possible find a representation A=XDX−1, where Dis a diagonal matrix.
1 1 1
1 1 1
1 1 1
,
1 2 1
0 3 1
0 5 −1
,3−1
1 1 ,1 1
−2 3
Problem 2. For each diagonalizable matrix from Problem 1 compute A6, find Bsuch
that B3=A, evaluate eAt.
Problem 3. Use the definition of the matrix exponential to compute eAt for each of the
following matrices:
1 1
0 1 ,1 1
−1−1
Problem 4. Find the general solution to Y0=AY . Solve the initial value problem.
a) A=
1 1 1
1 1 1
1 1 1
, Y (0) =
1
0
2
b) A=1 1
−2 3 , Y (0) = −1
2
