Chapter 2 – Linear Transformations Outline 2.1 Introduction to Linear Transformations and Their Inverses 2.2 Linear Transformations in Geometry 2.3 The Inverse of a Linear Transformation 2.4 Matrix Products

2.1 Introduction to Linear Transformations and Their Inverses The matrix A is called the coefficient matrix of the transformation. is called a linear transformation. : the decoding transformation. : the inverse of the coding transformation. Since the decoding transformation is the inverse of the coding transformation , we say that the matrix B is the inverse of the matrix A. We can write this as .

Invertible No all linear transformations are invertible. Suppose with matrix , the solutions are Because this system does not have a unique solution, it is impossible to recover the actual position from the encoded position. The coding matrix A is noninvertible. Consider two sets X and Y. A function T from X to Y is a rule that associates with each element x of X a unique element y of Y. The set X is called the domain of the function, and Y is its codomain. We will sometimes refer to x as the input of the function and to y as its output.

Linear Transformations (Definition 2.1.1) A function T from Rn to Rm is called a linear transformation if there is an m ×n matrix A such that for all in Rn. A linear transformation is a special kind of function. The identity transformation from Rn to Rn: all entries on the main diagonal are 1, and all other entries are 0. This matrix is called the identity matrix and is denoted by In. , where . The output vector is obtained from by rotating through an angle of 90o in the counterclockwise.

(Fact 2.1.2) The Column of the Matrix of a Linear Transformation (Fact 2.1.2) Consider a linear transformation T from Rm to Rn. Then, the matrix of T is To justify this result, we have then The vectors in Rm are sometimes referred to as the standard vectors in Rm. The standard vectors in R3 are often denoted by

(Fact 2.1.3) Linear Transformations (Fact 2,1,3) A transformation T from Rn to Rm is linear if (and only if) , for all , in Rn, and , for all in Rn and all scalars k.

2.2 Linear Transformations in Geometry (Example 1) Consider the matrices Show the effect of each of these matrices on our standard letter L, and describe each transformation in words.

Scalings For any positive constant k, the matrix defines a scaling by k, since This is a dilation (or enlargement) if k exceeds 1, and it is a contraction (or shrinking) for values of k between 0 and 1.

Projections Consider a line L in the plane, running through the origin. Any vector in R2 can be written uniquely as where is parallel to line L, and is perpendicular to L. The transformation from R2 to R2 is called the projection of onto L, often denoted by :

Example 2 (Example 2) Find the matrix A of the projection onto the line L spanned by

Definition 2.2.1 Projections (Definition 2.2.1) Consider a line L in the coordinate plane, running through the origin. Any vector in R2 can be written uniquely as where is parallel to line L, and is perpendicular to L. The transformation from R2 to R2 is called the projection of onto L, often denoted by . If is a unit vector parallel to L, then The transformation is linear, with matrix

Reflections

Definition 2.2.2 Reflections (Definition 2.2.2) Consider a line L in the coordinate plane, running through the origin, and let be a vector in R2. The linear transformation is called the reflection of about L, often denoted by : We have a formula relating to : The matrix of T is of the form , where a2+b2=1.

Projections and Reflections in Space Let be the plane through the origin perpendicular to L; note that the vector will be parallel to . We can give formulas for the orthogonal projection onto V, as well as for the reflections about V and L, in terms of the projection onto L:

Example 3 (Example 3) Let V be the plane defined by 2x1+x2-2x3=0, and let . Find

Fact 2.2.3 Rotations (Fact 2.2.3) The matrix of a counterclockwise rotation in R2 through an angle θ is Note that this matrix is of the form , where a2+b2=1.

Rotations The matrix of a counterclockwise rotation through an angle a is The matrix of a counterclockwise rotation through an angle of is

Example 4 &5 (Example 4) The matrix of a counterclockwise rotation through π/6 (or 30o) is (Example 5) Examine how the linear transformation affects our standard letter L. Here a and b are arbitrary constants.

Fact 2.2.4 Rotations Combined with a Scaling (Fact 2.2.4) A matrix of the form represents a rotation combined with a scaling. More precisely, if r and θ are the polar coordinates of vector , then represents a rotation through θ combined with a scaling by r.

Rotation-Dilations

Shears Let L be a line in R2. A linear transformation T from R2 to R2 is called a shear parallel to L if , for all vector on L, and is parallel to L for all vectors in R2.

Fact 2.2.5 Horizontal and Vertical Shears (Fact 2.2.5) The matrix of a horizontal shear is of the form , and the matrix of a vertical shear is of the form , where k is an arbitrary constant.

Shears (II)

2.3 The Inverse of Linear Transformation (Definition 2.3.1) A function T from X to Y is called invertible if the equation T(x)=y has a unique solution x in X for each y in Y. If a function T is invertible, then so is T-1, and (T-1)-1=T.

Invertible Consider the linear transformation from Rn to Rm given by where A is an matrix m<n The system has either no solutions or infinitely many solutions. The transformation is noninvertible. m=n The system has a unique solution if and only if rref(A)=In. Therefore, the transformation is invertible if and only if reff(A)=In, or equivalently, if rank(A)=n. m>n The transformation is noninvertible since the system is inconsistent.

Invertible Matrices (Definition 2.3.2) A matrix A is called invertible if the linear transformation is invertible. The matrix of the inverse transformation is denoted by A-1. If the transformation is invertible, its inverse is . (Fact 2.3.3) An matrix A is invertible if and only if A is a square matrix (i.e., m=n), and rref(A)=In. (Fact 2.3.4) Let A be an matrix Consider a vector in Rn. If A is invertible, then the system has the unique solution . If A is noninvertible, then the system has infinitely many solutions or none. Consider the special case when . The system has as a solution. If A is invertible, then this is the only solution. If A is noninvertible, then there are infinitely many other solutions.

Example 1 (Example 1) Is the matrix A invertible?

Finding the Inverse of a Matrix If a matrix A is invertible, how can we find the inverse matrix A-1?

Finding the Inverse of a Matrix (Fact 2.3.5) To find the inverse of an n×n matrix A, form the n× (2n) matrix [A|In] and compute rref[A|In]. If rref[A|In] is of the form [In|B], then A is invertible, and A-1=B. If rref[A|In] is of another form (i.e., its left half fails to be In), then A is not invertible. Note that the left half of rref[A|In] is rref(A).

Inverse and Determinant of a 2×2 Matrix The 2×2 matrix is invertible if (and only if) ad-bc≠0. Quantity ad-bc is called the determinant of A, written det(A): If is invertible, then

2.4 Matrix Products The composite of two functions: The composite of the functions y=sin(x) and z=cos(y) is z=cos(sin(x)).

Matrix Multiplication

Definition 2.4.1 Matrix Multiplication Let B be an n×p matrix and A a q×m matrix. The product BA is defined if (and only if) p=q. If B is an n×p matrix and A a p×m matrix, then the product BA is defined as the matrix of the linear transformation . This means that , for all in Rm. The product BA is an n×m matrix.

Fact 2.4.2 The Columns of the Matrix Product (Fact 2.4.2) Let B be n×p matrix and A a p×m matrix with columns Then, the product BA is To find BA, we can multiply B with the columns of A and combine the resulting vectors.

Fact 2.4.3 (Fact 2.4.3) Matrix multiplication is noncommutative. AB≠BA, in general. However, at times it does happen that AB=BA; then we say that the matrices A and B commute.

Fact 2.4.4 The Entries of the Matrix Product (Fact 2.4.4) Let B be an n×p matrix and A a p×m matrix. The ijth entry of BA is the dot product of the ith row of B with the jth column of A. Is the n×m matrix whose ijth entry is

Example 1 (Example 1)

Fact 2.4.5 Multiplying with the Inverse (Fact 2.4.5) For an invertible n×n matrix A,

Fact 2.4.6 Multiplying with the Identity Matrix (Fact 2.4.6) For an n×m matrix A,

Fact 2.4.7 Matrix Multiplication is Associative (Fact 2.4.7) Matrix multiplication is associative (AB)C=A(BC) We can simply write ABC for the product (AB)C=A(BC).

Fact 2.4.8 The Inverse of a Product of Matrices (Fact 2.4.8) If A and B are invertible n×n matrices, then BA is invertible as well, and (BA)-1=A-1B-1. Pay attention to the order of the matrices. (Order matters!)

Fact 2.4.9 A Criterion for Invertibility (Fact 2.4.9) Let A and B be two n×n matrices such that BA=In Then, A and B are both invertible, A-1=B and B-1=A, and AB=In.

Example 2 (Example 2) Suppose A, B, and C are three n×n matrices and ABC=In. Show that B is invertible, and express B-1 in terms of A and C.

Fact 2.4.10 Distributive Property for Matrices (Fact 2.4.10) If A and B are n×p matrices, and C and D are p×m matrices, then A(C+D)=AC+AD, and (A+B)C=AC+BC.

Fact 2.4.11 (Fact 2.4.11) If A is an n×p matrix, B is a p×m matrix, and k is a scaler, then (kA)B=A(kB)=k(AB).

Fact 2.4.12 Multiplying Partitioned Matrices (Fact 2.4.12) Partitioned matrices can be multiplied as though the submatrices were scalars (i.e., using the formula in Fact 2.4.4): is the partitioned matrix whose ijth “entry” is the matrix provided that all the products AikBkj are defined.

Example 3 (Example 3)

Example 4 (Example 4) Let A be a partitioned matrix where A11 is an n×n matrix, A22 is an m×m matrix, and A12 is an n×m matrix. For which choices of A11, A12, and A22 is A invertible? If A is invertible, what is A-1 (in terms of A11, A12, A22)?

Example 5 (Example 5)