Graded Assignment 6 (modules 11 and 12)
Material on basis and dimension
Problem 1: Consider the set of vectors in R 3 of the form
 a  b, b, 2a  5b 
(a) Prove that this set is a subspace of R 3 .
(b) Find a basis for the subspace.
(c) Is the vector u  4, 1,1  in the subspace? If so, express u as a linear combination of the
basis vectors for the subspace.
(d) Give the dimension of the subspace.
(e) Sketch the subspace.
Problem 2: Determine whether the following sets of vectors form a basis for R 3 . Justify your answer
(you have various theorems at your disposal, and in some cases you may be able to answer with doing
any calculation). If the set is not a basis for R 3 , explain whether it fails on span, independence, or both.
(a)  1,1,1 
 2,3, 4   5, 4,3   2,1, 4 
(b)  1,1,1 
 2, 4, 1 
(c)  1,1,1 
 2, 4, 1   2, 6, 4 
(d)  1,1,1 
 2, 4, 1   2, 6, 8 
Problem 3: Do the functions
f ( x)  x 2  6
g ( x)  x  5
h( x)  2 x 2  1
form a basis for P 2 (the space of polynomials of degree less than or equal to two)? Support your
answer.
Problem 4: For the matrix
1 3 1
0 3 2
A
1 1 1

 3 11 3
5
1 
1

9
(a) Give bases for the row and column spaces of A . What are the dimensions of these spaces?
(b) What is rank(A)?
Problem 5: For each of the given systems of equations, determine if the system has
unique/no/infinitely many solutions, and explain why by discussing the rank of the coefficient matrix vs.
the rank of the augmented matrix for the system.
(a) System 1
3x1  x2  x3

1
2 x1  3x2  x3

1
x2  x3
 2
3x1  x2  x3
 1
2 x1  3x2  x3
 1
(b) System 2
x1  2 x2
 3
Linear transformations
Problem 6: For each of the transformations given, either (1) prove that it's linear, or (2) give a
counterexample showing that it's non-linear.
(a) T : R 2  R 3 , T ( x, y )  x  y, y  3, 2 x  .
(b) T : R 2  R 2 , T ( x, y )  2 x  3 y,5 y  x  .
(c) T : R 3  R 2 , T ( x, y, z )  xz, yz 
Problem 7: Write the matrix representation of the transformation
T : R3  R 2 , T ( x, y, z )  2 x  3 y  z,5z  x 
and find the image of u  1, 0,1  under the transformation.
Problem 8: Give the matrix representation of the transformation in the plane
T  T3 T2 T1
where
T1 is a rotation through an angle of 30 degrees.
T2 is a reflection across the origin.
T3 is a contraction by a scale factor of
1
2
Apply the composite transformation to the unit square. Verify visually that the transformed square has
been rotated, reflected, and shrunk as indicated above.
Problem 9: Determine the kernel and range for the transformation
T : R3  R 2 , T ( x, y, z )  2 x  3z, y 
Give a basis for each, and state the dimension. Verify that the rank/nullity theorem holds.
Problem 10: Become a computer animator! Here's a picture:
Wreck the house. Go ahead, warp it, skew it, flatten it ... to accomplish this, you need to




Record the coordinates of all the vertices.
Enter them as columns in a matrix.
Invent a 2  2 transformation matrix. Make sure it is non-singular.
And transform the vertices. (One of the lectures covers how to transform a set of vertices in
one operation; be sure to find that)
Experiment with a few transformation matrices; see what you get. These don't have to be the
rotation/dilation/reflection transformations - you can invent any non-singular matrix and see what it
does.
Please submit TWO of your favorite transformed houses by including
(a) A sketch (you can do this by hand, on graph paper)
(b) The SciLab code used to do the transformations (so I can clearly see what you put down for the
original vertices, what your transformation matrix is, and what your new vertices are).