Linear Transformations and Their Matrices

Linear transformations and their matrices In older linear algebra courses, linear transformations were introduced before matrices. This geometric approach to linear algebra initially avoids the need for coordinates. But eventually there must be coordinates and matrices when the need for computation arises. Without coordinates (no matrix) Example 1: Projection We can describe a projection as a linear transformation T which takes every vec­ tor in R2 into another vector in R2 . In other words, T : R2 −→ R2 . The rule for this mapping is that every vector v is projected onto a vector T (v) on the line of the projection. Projection is a linear transformation. Definition of linear A transformation T is linear if: T (v + w) = T (v) + T (w) and T (cv) = cT (v) for all vectors v and w and for all scalars c. Equivalently, T (cv + dw) = cT (v) + dT (w) for all vectors v and w and scalars c and d. It’s worth noticing that T (0) = 0, because if not it couldn’t be true that T (c0) = cT (0). Non-example 1: Shift the whole plane Consider the transformation T (v) = v + v0 that shifts every vector in the plane by adding some fixed vector v0 to it. This is not a linear transformation because T (2v) = 2v + v0 �= 2T (v). Non-example 2: T (v) = ||v|| The transformation T (v) = ||v|| that takes any vector to its length is not a linear transformation because T (cv) �= cT (v) if c < 0. We’re not going to study transformations that aren’t linear. From here on, we’ll only use T to stand for linear transformations. Example 2: Rotation by 45◦ This transformation T : R2 −→ R2 takes an input vector v and outputs the vector T (v) that comes from rotating v counterclockwise by 45◦ about the ori­ gin. Note that we can describe this and see that it’s linear without using any coordinates. The big picture One advantage of describing transformations geometrically is that it helps us to see the big picture, as opposed to focusing on the transformation’s effect on a single point. We can quickly see how rotation by 45◦ will transform a picture of a house in the plane. If the transformation was described in terms of a matrix rather than as a rotation, it would be harder to guess what the house would be mapped to. Frequently, the best way to understand a linear transformation is to find the matrix that lies behind the transformation. To do this, we have to choose a basis and bring in coordinates. With coordinates (matrix!) All of the linear transformations we’ve discussed above can be described in terms of matrices. In a sense, linear transformations are an abstract description of multiplication by a matrix, as in the following example. Example 3: T (v) = Av Given a matrix A, define T (v) = Av. This is a linear transformation: A(v + w) = A(v) + A(w) and A(cv) = cA(v). Example 4 � Suppose A = 1 0 0 −1 � . How would we describe the transformation T (v) = Av geometrically? When we multiply A by a vector v in R2 , the x component of the vector is unchanged and the sign of the y component of the vector is reversed. The transformation v �→ Av reflects the xy-plane across the x axis. Example 5 How could we find a linear transformation T : R3 −→ R2 that takes three dimensional space to two dimensional space? Choose any 2 by 3 matrix A and define T (v) = Av. Describing T (v) How much information do we need about T to to determine T (v) for all v? If we know how T transforms a single vector v1 , we can use the fact that T is a linear transformation to calculate T (cv1 ) for any scalar c. If we know T (v1 ) and T (v2 ) for two independent vectors v1 and v2 , we can predict how T will transform any vector cv1 + dv2 in the plane spanned by v1 and v2 . If we wish to know T (v) for all vectors v in Rn , we just need to know T (v1 ), T (v2 ), ..., T (vn ) for any basis v1 , v2 , ..., vn of the input space. This is because any v in the input space can be written as a linear combination of basis vectors, and we know that T is linear: v = c1 v1 + c2 v2 + · · · + c n v n T ( v ) = c1 T ( v1 ) + c2 T ( v2 ) + · · · + c n T ( v n ). This is how we get from a (coordinate-free) linear transformation to a (co­ ordinate based) matrix; the ci are our coordinates. Once we’ve chosen a basis, every vector v in the space can be written as a combination of basis vectors in exactly one way. The coefficients of those vectors are the coordinates of v in that basis. Coordinates come from a basis; changing the basis changes the coordinates of vectors in the space. We may not use the standard basis all the time – we sometimes want to use a basis of eigenvectors or some other basis. The matrix of a linear transformation Given a linear transformation T, how do we construct a matrix A that repre­ sents it? First, we have to choose two bases, say v1 , v2 , ..., vn of Rn to give coordi­ nates to the input vectors and w1 , w2 , ..., wm of Rm to give coordinates to the output vectors. We want to find a matrix A so that T (v) = Av, where v and Av get their coordinates from these bases. The first column of A consists of the coefficients a11 , a21 , ..., a1m of T (v1 ) = a11 w1 + a21 w2 + · · · + a1m wm . The entries of column i of the matrix A are de­ termined by T (vi ) = a1i w1 + a2i w2 + · · · + a1i wm . Because we’ve guaranteed that T (vi ) = Avi for each basis vector vi and because T is linear, we know that T (v) = Av for all vectors v in the input space. In the example of the projection matrix, n = m = 2. The transformation T projects every vector in the plane onto a line. In this example, it makes sense to use the same basis for the input and the output. To make our calculations as simple as possible, we’ll choose v1 to be a unit vector on the line of projection and v2 to be a unit vector perpendicular to v1 . Then T ( c1 v1 + c2 v2 ) = c1 v1 + 0 � and the matrix of the projection transformation is just A = � Av = 1 0 0 0 �� c1 c2 � � = c1 0 1 0 0 0 � . � . This is a nice matrix! If our chosen basis consists of eigenvectors then the matrix of the transformation will be the diagonal matrix Λ with eigenvalues on the diagonal. To see how important the choice of basis is, let’s use the standard basis for the linear transformation� that� projects the plane �onto�a line at a 45◦ angle. If 1 0 we choose v1 = w1 = and v2 = w2 = , we get the projection 0 1 � � aa T 1/2 1/2 matrix P = T = . We can check by graphing that this is the 1/2 1/2 a a correct matrix, but calculating P directly is more difficult for this basis than it was with a basis of eigenvectors. Example 6: T = d dx Let T be a transformation that takes the derivative: T (c1 + c2 x + c3 x2 ) = c2 + 2c3 x. (1) The input space is the three dimensional space of quadratic polynomials c1 + c2 x + c3 x2 with basis v1 = 1, v2 = x and v3 = x2 . The output space is a two dimensional subspace of the input space with basis w1 = v1 = 1 and w2 = v2 = x. � � This is a linear transformation! So we can find A = 0 0 the transformation (1) as a matrix multiplication (2): ⎛⎡ ⎤⎞ ⎡ ⎤ � � c1 c1 c2 T ⎝⎣ c2 ⎦⎠ = A ⎣ c2 ⎦ = . 2c3 c3 c3 1 0 0 2 and write (2) Conclusion For any linear transformation T we can find a matrix A so that T (v) = Av. If the transformation is invertible, the inverse transformation has the matrix A−1 . The product of two transformations T1 : v �→ A1 v and T2 : w �→ A2 w corresponds to the product A2 A1 of their matrices. This is where matrix multi­ plication came from!