Linear Transformations

Transcription

1 Linear Transformations Systems of linear equations, with matrix form Ax = b, are often usefully analyzed by viewing the equation as the problem that asks for an unknown input x for a function that produces a known output b. The rule for this function is the one that takes vector inputs x R n and returns vector outputs Ax R m. We call such functions with vector inputs and vector outputs transformations (of Euclidean spaces). Using standard notation for describing functions, we can refer to such a transformation as a function, or map, of the form T:R n R m. Here, T is the name of the transformation that carries vector inputs from R n to vector outputs from R m according to some well-defined rule. A particularly simple example of a transformation is the identity map T :R n R n with formula T(x) = x. In the context of systems of linear equations (the one of prime interest to us), we could define T by the formula T(x) = Ax where A is some m n matrix. See Example 1, p. 74, for details, and study the marginal diagrams there for a glimpse at how one might form a mental picture of such a transformation.

2 Examples 2, 3, 4, and 5 on pp illustrate that much geometrical information is captured by the behavior of certain transformations of the form T(x) = Ax. It is not coincidence that these transformations obey the simple properties T(u + v) = T(u) +T(v) for all inputs u, v; T(cu) = ct(u) for all inputs u and scalars c. Any transformation of Euclidean spaces that satisfies these two properties is called a linear transformation. The defining properties can be interpreted as saying that linear transformations preserve vector additions and scalar multiplications. Because of the algebraic properties of the matrixvector product, it is clear that all transformations of the form T(x) = Ax are automatically linear transformations. But as we shall soon see, the converse statement is also true: every linear transformation has a matrix form T(x) = Ax! For instance, check that the identity map T(x) = x is a linear transformation. It can be represented in terms of the matrix-vector product formula T(x) = I n x, involving the n n identity matrix: I n =

3 Let s investigate some other important properties of linear transformations: Theorem Any linear transformation T:R n R m takes the zero vector in R n to the zero vector in R m. Proof T(0) = T(0 + 0) = T(0) +T(0) T(0) = 0. // Theorem The transformation T:R n R m is linear if and only if for all vectors u,v R n and all scalars c, d, the relation T(cu +dv) = ct(u)+ dt(v) holds. Proof If T is a linear transformation, then, using the defining properties, we have T(cu +dv) = T(cu) +T(dv) = ct(u) + dt(v). Conversely, if T is a transformation for which T(cu +dv) = ct(u)+ dt(v) holds, then setting c = d = 1 shows that T(u + v) = T(u) +T(v) for all u and v in R n ; and setting d = 0 shows that T(cu) = ct(u) for every choice of c and u. So T must be linear. //

4 Repeated application of the property T(cu +dv) = ct(u)+ dt(v), shows that if T is a linear transformation, then for any collection of vectors v 1,v 2,,v k and associated scalars c 1,c 2,,c k, T(c 1 v 1 +c 2 v 2 + +c k v k ) = c 1 T(v 1 ) +c 2 T(v 2 )+ + c k T(v k ) That is, T carries any linear combination of a set v 1,v 2,,v k of vectors in R n to the same linear combination of their images T(v 1 ),T(v 2 ),,T(v k ) in R m. This is often referred to as the superposition principle. We are now in a position to prove the theorem alluded to earlier: Theorem Any linear transformation T:R n R m has an associated m n matrix A for which T(x) = Ax. More specifically, A = [ T(e 1 ) T(e 2 ) T(e n )] is the matrix whose jth column is the image T(e j ) of the vector e j which is the jth column of the n n identity matrix.

5 Proof The identity matrix I n satisfies x = I n x = [ e 1 e 2 e n ]x = x 1 e 1 + x 2 e x n e n So, by the linearity of T, T(x) = T(x 1 e 1 + x 2 e x n e n ) = x 1 T(e 1 ) + x 2 T(e 2 )+ + x n T(e n ) x 1 = [ T(e 1 ) T(e 2 ) T(e n ) x ] 2 = Ax. // x n The matrix A obtained by applying this theorem to a linear transformation T is called its standard matrix. For instance, pp present standard matrices for the linear transformations from R 2 to R 2 which represent reflections across lines, reflection through a point (the origin), dilations and shears, and projections onto certain lines.

6 It is significant to note that the geometric transformations in Tables 1-3 (pp ) are one-toone as functions, and they map R 2 onto R 2. In contrast, the projection maps in Table 4 (p. 87) are neither one-to-one nor onto. The properties of being one-to-one and onto are related to ideas we have explored earlier; this is spelled out in the following two theorems: Theorem The linear transformation T:R n R m is one-to-one if and only if the zero vector in R n is the only vector that is mapped by T to the zero vector in R m, i.e., T(x) = 0 has only the trivial solution. Proof Since T is a linear transformation, T(0) = 0. So, if T is one-to-one, T(x) = 0 can have only the trivial solution x = 0. Conversely, suppose T is a linear transformation for which T(x) = 0 has only the trivial solution. Then, if u and v are vectors in R n for which T(u) = T(v), it follows that T(u v) = T(u) T(v) = 0 from which we conclude that u v = 0, or u = v. Therefore, T is one-to-one. //

7 Theorem Suppose the linear transformation T:R n R m has standard matrix A. Then (1) T is one-to-one if and only if the columns of A are linearly independent, that is, if and only if A has a pivot entry in every column; and (2) T is onto if and only if the columns of A span R m, that is, if and only if A has a pivot entry in every row. Proof (1) is an immediate consequence of the previous theorem, since we know that the columns of A are linearly independent if and only if Ax = 0 has only the trivial solution, which happens if and only if the solution to the homogeneous system has no free variables. To prove (2), recall the theorem that says that the columns of A span R m if and only if the equation Ax = b is consistent for every b R n. But this holds if and only if T(x) = Ax is an onto function. In particular, if A has a pivot entry in every row, we are assured that the system Ax = b always has a solution; but conversely, if Ax = b has a solution x for every possible b R n, then every b is a linear combination of the columns of A. //