(15 points)
The matrices A and B below are row equivalent (you do not need to
check this fact).
a) What is the rank of A?
b) Find a basis for the null space Null(A) of A.
c) Find a basis for the column space of A.
d) Find a basis for the row space of A.
(6 points)
The system has a 2-dimensional space of solutions
and the size of the matrix A is .
What is the dimension of
(a) the Null space of A?
(b) the Column space of A?
(c) the Row space of A?
Justify your answers!
(15 points)
Show that the characteristic polynomial of the matrix
is .
Find a basis of consisting of eigenvectors of A.
Find an invertible matrix P and a diagonal matrix D such that
the matrix A above satisfies
(12 points)
Determine for which of the following matrices A below there exists an
invertible matrix P (with real entries)
such that is a diagonal matrix.
You do not need to find P.
Justify your answers!
(20 points)
Let W be the plane in spanned by
and
Note: Parts 5a,
5b are mutually independent and are not needed for
doing parts 5c, 5d,
5e.
Find the distance between the two points and
in .
Find a vector of length 1 which is orthogonal to W.
Find the projection of to the line spanned by .
Write as the sum of a vector parallel to and a
vector orthogonal to .
Find an orthogonal basis for W.
Find an orthogonal matrix U, such that the corresponding
linear transformation from to
takes the axis to the line spanned by and the ,
coordinate plan to W.
Hint: Use parts 5b and 5d.
(16 points)
Let W be the plane in spanned by
and
Find the projection of
to W.
Find the distance from b to W.
Find a least square solution of the equation Ax=b, where
is the
matrix with columns and . I.e., find a vector x
in which minimizes the length .
Find the coefficients , of the line which best
fits the three points
, , in the x,y plane.
The line should minimize the sum
.
Justify your answer!
(16 points)
The vectors
and
are eigenvectors of the matrix
.
The eigenvalue of is:
The eigenvalue of is:
Find the coordinates of
in the basis .
Compute .
As n gets larger, the vector approaches Justify your answer.