PLAN ESTRATÉGICO DE SUBVENCIONES
PROGRAMA 1311: ACTUACIONES DE RELACIONES EXTERIORES Objetivos estratégicos:
dim W = m . Assume T V, W) and let r = dim T(V) denote the rank of T. Then there exists a basis (e, , . . . , e,) for V and a basis . . . , W such that
(2.14) for r,
and
(2.15) f o r
Therefore, the matrix of T relative to these bases has all entries zero except for the r diagonal entries
= = = = 1 .
Proof. First we construct a basis for W. Since T(V) is a of W with dim T(V) = r, the space T(V) has a basis of r elements in say . . . , By Theorem 1.7, these elements form a subset of some basis for W. Therefore we can adjoin elements . . . ,
so that
(2.16) . .
Construction of a matrix representation in diagonal form 49 Now we construct a basis for Each of the first r elements in (2.16) is the image of at least one element in Choose one such element in and call it . Then = for
r so (2.14) is satisfied. Now let be the dimension of the null space N(T). By Theorem 2.3 we have n = k + Since dim N(T) k, the space N(T) has a basis consisting of k elements in which we designate as , . . . , For each of these elements, Equation (2.15) is satisfied. Therefore, to the proof, we must show that the ordered set
(2.17) . . . , . . . ,
is a basis Since dim = n = r + k, we need only show that these elements are independent. Suppose that some linear combination of is zero, say
(2.18)
Applying and using Equations (2.14) and (2.19, we find that
But are independent, and hence 0. Therefore, the first terms in (2.18) zero, so (2.18) reduces to
, , are independent since they form a basis for N(T), and hence =
. 0. Therefore, all the in (2.18) are zero, the elements in (2.17) form a basis for This completes the proof.
EXAMPLE. We refer to Example 2 of Section 2.10, where is the differentiation operator which maps the space of polynomials of degree 3 into the space W of polynomials of degree In this example, the range T(V) = W, so T has rank 3. Applying the method used to prove Theorem 2.14, we choose any basis for W, for example the basis (1, A set of polynomials in which map onto these elements is given by (x, We extend this set to get a basis for by adjoining the constant polynomial which is a basis for the null space of D. Therefore, if we use the basis 1) for and the basis (1, x, for W, the corresponding matrix representation for D has the diagonal form
1 0 0 0
0 1 0 0 .
50 Linear transformations and matrices
2.12 Exercises
In all exercises involving the vector space , the usual basis of unit coordinate vectors is to be chosen unless another basis is specifically mentioned. In exercises concerned with the matrix of a linear transformation T: where = we take the same basis in both and W unless
another choice is indicated.
1. Determine the matrix of each of the following linear transformations of into : (a) the identity transformation,
(b) the zero transformation,
(c) multiplication by a fixed scalar c.
2. Determine the matrix for each of the following projections.
(a) T: where , = .
(b) T:
=
(c) T:
3. A linear transformation T: maps the basis vectors andj as follows:
T(i) = i j,
(a) Compute 4j) and 4j) in terms of and j. (b) Determine the matrix of T and of
(c) Solve part (b) if the basis (i, j) is replaced by , where j, = 3i + j. 4. A linear transformation T: is defined as follows: Each vector (x, is reflected in
the y-axis and then doubled in length to yield Determine the matrix of T and of Let T: be a linear transformation such that
T(k) T ( j + k ) T ( i + j + k ) = j - k .
(a) Compute 2j 3k) and determine the nullity and rank of T.
(b) Determine the matrix of T.
6. For the linear transformation in Exercise 5, choose both bases to be (e,, where = (2, 3, = (1, 0, = (0, 1, and determine the matrix of T relative to the new bases.
7. A linear transformation T: maps the basis vectors as follows: T(i) = (0, T(j) = = - 1 ) .
(a) Compute -j + k) and determine the nullity and rank of T.
(b) Determine the matrix of T.
(c) Use the basis (i, j, k) in and the basis , in where = (1, 1), = Determine the matrix of T relative to these bases.
(d) Find bases , for and , for relative to which the matrix of Twill be in diagonal form.
8. A linear transformation maps the basis vectors as follows: T(i) = 0, T(j) = 1).
(a) Compute 3j) and determine the nullity and rank of T.
(b) Determine the matrix of T.
(c) Find bases (e,, e,) for and (w,, for relative to which the matrix of Twill be in diagonal form.
9. Solve Exercise 8 if T(i) = 0, and T(j) = (1, 1).
10. Let W spaces, each with dimension 2 and each with basis , Let T: W be a linear transformation such that + = + =
(a) Compute and determine the nullity and rank of (b) Determine the matrix of T relative to the given basis.
Linear spaces of matrices 5 1 (c) Use the basis , e,) for V and find a new basis of the form + + be,) for W, relative to which the matrix of Twill be in diagonal form.
In the linear space of all real-valued functions, each of the following sets is independent and spans a finite-dimensional Use the given set as a basis for and let D: V be the operator. In each case, find the matrix of D and of relative to this choice of basis.
11. (sin x, cos x). 15. (-cos x, sin x).
12. (1, 16. (sin x, cos x sin x, x cos x).
13. (1, I x, I + 17. sin x, cos x).
14. 18. sin 3x, cos 3x).
19. Choose the basis (1, x, in the linear space V of all real polynomials of degree Let D denote the differentiation operator and let T: V be the linear transformation which onto Relative to the given basis, determine the matrix of each of the followingtransformations: (a) T; DT; TD; TD DT; 20. Refer to Exercise 19. Let W be the image of Vunder TD. Find bases for Vand for W relative
to which the matrix of TD is in diagonal form.
2.13 Linear spaces of matrices
We have seen how matrices arise in a natural way as representations of linear trans- formations. Matrices can also be considered as objects existing in their own right, without necessarily being connected to linear transformations. As such, they form another class of mathematical objects on which algebraic operations can be defined. The connection with linear transformations serves as motivation for these definitions, but this connection will be ignored for the moment.
Let and be two positive integers, and let be the set of all pairs of integers such that 1 1 Any function A whose domain is is called an m x n matrix. The value j) is called the or ij-element of the matrix and will be denoted also by . It is customary to display all the function values in a rectangular array consisting of m rows and n columns, as follows:
. .
The elements may be arbitrary objects of any kind. Usually they will be real or complex numbers, but sometimes it is convenient to consider matrices whose elements are other objects, for example, functions. We also denote matrices by the more compact notation
A = or A =
If m = n , the is said to be a square matrix. A x n matrix is called a row matrix; an m x 1 matrix is called a column matrix.
52 Linear transformations and matrices
Two functions are equal if and only if they have the same domain and take the same function value at each element in the domain. Since matrices are functions, two matrices A = and are equal if and only if they have the same number of rows, the same number of columns, and equal entries = for each pair
Now we assume the entries are numbers (real or complex) and we define addition of matrices and multiplication by scalars by the same method used for any real- or
valued functions.
DEFINITION. If A and B = are two m x n matrices and is any scalar, we define matrices A + B and as follows:
=
The sum is defined only when A and B have the same size.
EXAMPLE. If
1 2 - 3
A = 0 4 I and B =
then we have
We define the zero matrix 0 to be the m x n matrix all of whose elements are 0. -With these definitions, it is a straightforward exercise to verify that the collection of all m x n matrices is a linear space. We denote this linear space by If the entries are real numbers, the space is a real linear space. If the entries are complex, is a complex linear space. It is also easy to prove that this space has dimension mn. In fact, a basis for
M consists of the mn matrices having one entry equal to 1 and all others equal to 0. the six matrices
form a basis for the set of all 2 x 3 matrices.
2.14 Isomorphism between linear transformations and matrices
We return now to the connection between matrices and linear transformations. Let and W be finite-dimensional linear spaces with dim V = n and dim W = m. Choose a basis (e,, . . . , e,) for V and a basis . . . , w,) for W. In this discussion, these bases are kept fixed. Let W) denote the linear space of all linear transformations of V into
W. If T W), let m(T) denote the matrix of T relative to the given bases. We recall