Fact: Every matrix transformation is a linear transformation, and vice versa.

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1 Linear Transformations Definition: A transformation (or mapping) T is linear if: (i) T (u + v) = T (u) + T (v) for all u, v in the domain of T ; (ii) T (cu) = ct (u) for all scalars c and all u in the domain of T Fact: Every matrix transformation is a linear transformation, and vice versa Therefore, if you can find a matrix to implement the transformation, it is a linear transformation Fact: If T is a linear transformation, then T () = and T (cu + dv) = ct (u) + dt (v) for all vectors u, v in the domain of T and all scalars c, d In general, T (c v + + c p v p ) = c T (v ) + + c p T (v p ) which is known as a superposition principle Example: Show that T (x, x 2, x 3 ) = (3x, 3, x 2 + x 3 ) is not a linear transformation Solution: Note that the given transformation can be written as T x x 2 = 3x 3 x 3 x 2 + x 3 If c is any scalar, then T (cx) = T cx cx 2 = 3cx 3 ct (x) cx 3 cx 2 + cx 3 since c cannot be factored out Thus, T is not a linear transformation

2 Example: Show that T (x, x 2, x 3 ) = (x +x 2, 3x 2 2x 3 ) is a linear transformation by finding a matrix A that implements the mapping Solution: To show that T is a linear transformation, it suffices to show that we can write the mapping as T (x) = Ax x + x T (x) = 2 3x 2 2x 3 = x + x 2 + x = x x }{{} x 3 A }{{} x Since there exists a matrix A such that T (x) = Ax for all vectors x in R 3, T is a linear transformation Note that you can approach this problem by verifying that both properties of a linear transformation are satisfied However, in many cases, the computation can be rather lengthy Sometimes, it is much easier to determine the standard matrix of the transformation Example: Let x = x, v = x 2 x 3 [ [ [ 2, v ] 2 =, and v 3] 3 = and let T : R 2] 3 R 2 be a linear transformation that maps x into x v + x 2 v 2 + x 3 v 3 Find a matrix A so that T (x) = Ax for any x Solution: The transformation that we are setting up is simply T (x) = v v 2 v 3 x, ie 2 T (x) = x x x 3 Definition: The j th column of the identity matrix is denoted by e j For example, if I = is the 3 3 identity matrix, then e =, e 2 =, e 3 =

3 Theorem: Let T : R n R m be a linear transformation Then there exists a unique matrix A such that T (x) = Ax for all x in R n In fact, A is an m n matrix whose j th column is the vector T (e j ): A = [ T (e ) T (e n ) ] Example: Suppose that T : R 2 R 2 such that T (e ) = the standard matrix for T and T (e 2 ) = [ ] Find Solution: By the above theorem, we have A = [ T (e ) T (e 2 ) ] = Example: Find the standard matrix of T : R 2 R 2 that reflects points through the horizontal (x ) axis, then reflects through the line x 2 = x Solution: Let x be any vector in R 2 Note that x is a linear combination of the vectors e and e 2, ie, x = x e + x 2 e 2 For convenience, let s express x as an ordered pair x, x 2 There are two things that we must do (in the following order): First, we need to reflect x about the x -axis Then, we need to rotate the resultant vector about the line x 2 = x The vector that we obtain is the vector that we will call b (See Figure ) Figure : Sequence of reflections from x to b

4 Reflecting x about the x -axis yields a vector of the form x, x 2 (which is the green vector in Figure ) Reflecting this vector about the line x 2 = x yields a vector of the form b = x 2, x (we simply interchange the coordinates) Therefore, the transformation is () x x2 T = x 2 x Now that we have T, we can use it to find the columns of the standard matrix A Note that ([ [ ([ T (e ) = T = and T (e ]) ] 2 ) = T = ]) Hence, A = [ T (e ) T (e 2 ) ] = The standard matrix A in this case is an example of a rotation matrix This particular matrix rotates any vector in R 2 by 9 In Figure, the angle between x and b is 9 Dilation and Contraction Operators If k is a nonnegative scalar, then the transformation defined by T (x) = kx on R 2 or R 3 is called a contraction with factor k if k and a dilation with factor k if k > The geometric effect of a contraction is to compress each vector by a factor of k, and the effect of a dilation is to stretch each vector by a factor of k (See Figure 2) A contraction compresses R 2 or R 3 uniformly toward the origin from all directions, and a dilation stretches R 2 or R 3 uniformly away from the origin in all directions Figure 2: A contraction when k (left) and a dilation when k > (right)

5 Rotation Operators An operator that rotates each vector in R 2 through a fixed angle θ is called a rotation operator on R 2 Let s look at an example Example: Define T : R 2 R 2 by x T (x) = x 2 }{{}}{{} A x Note that the matrix A is the same matrix that we found [ in the previous example Let s play around with this transformation If we plug in x =, we obtain ] ( [ ) T = ] [ ] Notice what the transformation did to the input vector It appears that the standard matrix rotated the input vector by 9 to yield the output[ vector Is this a coincidence? Let s plug in a different vector and see what happens If x =, then ] ( [ ) T = ] The angle between the input vector and output vector is 9 Let s look at the standard matrix A and determine why it rotates any vector by 9 Note that cos(9 ) sin(9 ) A = = sin(9 ) cos(9 ) Should we be convinced that if we wish to rotate any vector by an angle θ that the matrix cos θ sin θ sin θ cos θ will do just that? Let s verify this To find equations relating x and b = T (x), let φ be the angle from the positive x -axis to x, and let r be the common length of x and b (See Figure 3) Then from basic trigonometry, we have and x = r cos φ, x 2 = r sin φ () b = r cos(θ + φ), b 2 = r sin(θ + φ) (2)

6 Figure 3: Geometric effect of the rotation operator on R 2 Using trigonometric identities on (2) yields Then from the equations in (), we have b = r cos θ cos φ r sin θ sin φ b 2 = r sin θ cos φ + r cos θ sin φ b = x cos θ x 2 sin θ b 2 = x sin θ + x 2 cos θ (3) which we can express in matrix-vector form as cos θ sin θ x = sin θ cos θ Note that the equations in (3) are linear, so T is a linear transformation Furthermore, it follows from these equations that the standard matrix for T is cos θ sin θ A = sin θ cos θ We have now established the fact that any vector in R 2 is rotated by an angle θ whenever the rotation matrix A acts on it by matrix-vector multiplication x 2 [ b b 2 ]

7 Definition: A mapping T : R n R m is said to be onto R m if each b in R m is the image of at least one x in R n Recall that the range of T is generally a subset of the codomain R m T is onto if and only if the range of T is exactly the same set as the codomain Definition: A mapping T : R n R m is said to be one-to-one if each b in R m is the image of at most one x in R n How do we determine if a linear transformation is one-to-one and/or onto? This question brings us to the following theorem Theorem: Let T : R n R m be a linear transformation and let A be the standard matrix for T Then a T maps R n onto R m if and only if the columns of A span R m b T is one-to-one if and only if the columns of A are linearly independent Example: Let T be a linear transformation whose standard matrix is A = (a) Does T map R 4 onto R 3? Solution: Since the matrix is already in echelon form, we can simply check if each row has a pivot Since every row does have a pivot, the columns of A span R 3 Thus, by the above theorem, T maps R 4 onto R 3 (b) Is T a one-to-one mapping? Solution: Since not every column of A is a pivot column, the columns of A are linearly dependent Thus, by the above theorem, T is not one-to-one Note that another way that we could establish linear dependence of the columns of A is to use the fact that the number of columns exceeds the number of rows