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  2b, b  a,5b 
a) Prove that this set is a subspace of R 3 by showing closure under addition and scalar
multiplication.
b) Find a basis for the subspace.
c) Is the vector w  8,5,15  in the subspace? If so, express w 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 partially answer with
doing any calculation. If the set is not a basis for R 3 , explain whether it fails on span, independence, or
both. You do need to show that you have considered both span and independence for each of these –
even if you know it fails one of those thing immediately by theorem, I still want you to discuss whether
is passes or fails the other as well.
(a)  1, 2,1 
 1,1, 2   1,5, 0 
(b)  3, 2,1 
 1, 2,1   4,1,3   1,1, 2 
(c)  1, 0, 2 
(d)  3, 2,1 
 1, 2,1   3,3, 2 
 4,1,3 
Problem 3: Do the functions
f ( x )  2 x 2  10
g ( x)  x2  5x  6
h( x )  x 2  5 x  4
2
form a basis for P (the space of polynomials of degree less than or equal to two)? Support your
answer.
Problem 4:
For the matrix
1 3 1 5 
 4 1 9 6 

A 
1 1 1 1 


1 0 2 1
a) Give bases for the row and column spaces of A . What are the dimensions of these spaces?
b) What is the value of 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  3 x2  x3

1
x2  x3
b) System 2:
 2
3 x1  x2  x3
 1
2 x1  3x2  x3
 1
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  )  4 x  3 y ,3 xy , x  2 y  .
(b) T : R 2  R 3 , T ( x, y )  4 y,  x  3 y, 2 x  .
(c) T : R 3  R 2 , T ( x, y, z )  x  z, 2 y  1  .
Problem 7: Write the matrix representation of the transformation
T : R3  R2 , T ( x, y, z )  2 x  5 y  z,3z  y 
and find the image of u  1,2,1  under the transformation.
Problem 8: Give the matrix representation of the transformation T in the plane, where T is defined by
T  T3 T2 T1
and where
T1 is a rotation through an angle of 60 degrees.
T2 is a reflection across the origin.
T3 is a contraction by a scale factor of
2
3
Apply the composite transformation to the unit square, giving the vertices of the new square under the
transformation. Verify visually (include sketch) 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  R2 , T ( x, y, z )  y,3z  4 x 
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).