5.4. EXERCISES 115
11. Let a be a fixed vector. The function Ta defined by Tav = a+v has the effect oftranslating all vectors by adding a. Show this is not a linear transformation. Explainwhy it is not possible to realize Ta in R3 by multiplying by a 3×3 matrix.
12. ↑In spite of Problem 11 we can represent both translations and linear transformationsby matrix multiplication at the expense of using higher dimensions. This is done bythe homogeneous coordinates. I will illustrate in R3 where most interest in this isfound. For each vector v = (v1,v2,v3)
T , consider the vector in R4 (v1,v2,v3,1)T .
What happens when you do1 0 0 a1
0 1 0 a2
0 0 1 a3
0 0 0 1
v1
v2
v3
1
?
Describe how to consider both linear transformations and translations all at once byforming appropriate 4×4 matrices.
13. You want to add(
1, 2, 3)
to every point in R3 and then rotate about the zaxis counter clockwise through an angle of 30◦. Find what happens to the point(
1, 1, 1).
14. Let P3 denote the set of real polynomials of degree no more than 3, defined on aninterval [a,b]. Show that P3 is a subspace of the vector space of all functions definedon this interval. Show that a basis for P3 is
{1,x,x2,x3
}. Now let D denote the
differentiation operator which sends a function to its derivative. Show D is a lineartransformation which sends P3 to P3. Find the matrix of this linear transformationwith respect to the given basis.
15. Generalize the above problem to Pn, the space of polynomials of degree no more thann with basis {1,x, · · · ,xn} .
16. If A is an n× n invertible matrix, show that AT is also and that in fact,(AT)−1
=(A−1
)T .
17. Suppose you have an invertible n×n matrix A. Consider the polynomialsp1 (x)
...pn (x)
= A
1...
xn−1
Show that these polynomials p1 (x) , · · · , pn (x) are a linearly independent set of func-tions.
18. Let the linear transformation be T = D2+1, defined as T f = f ′′+ f . Find the matrixof this linear transformation with respect to the given basis
{1,x,x2,x3
}.