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Department of Mathematics
2013 Semester 1 MATHS 208 – Assignment 2 Due: 4pm, Tuesday 14th May
Hand your assignment in to the correct box in the Basement of Building 303, before the due date
.
Please use a Department of Mathematics Cover Sheet, available from the Basement of Building 303.
Show all working and note that late assignments will not be marked.
This assignment was written by Julia Novak (j.novak@auckland.ac.nz). The tutors in the Mathematics
Assistance Room (on the Ground Floor of Building 303) can give you help and advice on the concepts
covered, but all work submitted must be your own.
1. Consider the matrix
A =
1 1 2 0
2 0 2 −
1
−1 0 −1 1
−2 −1 −3 0
and the vector b =
1
2
−2
−1
.
(a) Construct the augmented matrix [ A | b ] and use elementary row operations to transform it to
reduced row echelon form.
(b) Find a basis for the column space of A.
(c) State the dimension of the column space of A.
(d) Find a basis for the nullspace of A.
(e) State the nullity of A.
(f) Find a particular solution to the linear system Ax = b.
(g) Find the general solution to the linear system Ax = b, expressing your answer in the form of a
vector plus a subspace.
[18 marks]
2. (a) Find the orthogonal projection of the vector u = (2, 3) on the vector v = (−2, 5).
(b) Verify that w = u − proj
v
u is perpendicular to v.
[4 marks]
3. Use the method of Gram-Schmidt to obtain an orthonormal basis for the column space of the matrix
A =
2 1 3
0 1 −2
2 1 1
0 1 0
.
[12 marks]
4. Find all least squares solutions of Ax = b, where A =
1 3
−2 −6
3 9
, b =
1
0
1
and confirm that all
the solutions have the same error vector (and hence the same least squares error).
Compute the least-squares error. [12 marks]
5. Identify all real values of a for the following vectors to form a linearly dependent set in R3?
v1 =
(
a,
−
1
2
, −
1
2
)
, v2 =
(
−
1
2
, a, −
1
2
)
, v3 =
(
−
1
2
, −
1
2
, a
)
.
[4 marks]
2013 Semester 1 MATHS 208 – Assignment 2 Page 1 of 1
