1 Linear Equations in Linear Algebra LINEAR INDEPENDENCE

LINEAR INDEPENDENCE § Definition: An indexed set of vectors {v 1, …, vp} in is said to be linearly independent if the vector equation has only the trivial solution. The set {v 1, …, vp} is said to be linearly dependent if there exist weights c 1, …, cp, not all zero, such that ----(1) © 2012 Pearson Education, Inc. Slide 1. 7 - 2

LINEAR INDEPENDENCE § Equation (1) is called a linear dependence relation among v 1, …, vp when the weights are not all zero. § An indexed set is linearly dependent if and only if it is not linearly independent. § Example 1: Let © 2012 Pearson Education, Inc. , , and . Slide 1. 7 - 3

LINEAR INDEPENDENCE § Substitute these values into equation (1) and obtain the equation below. § This is one (out of infinitely many) possible linear dependence relations among v 1, v 2, and v 3. © 2012 Pearson Education, Inc. Slide 1. 7 - 4

LINEAR TRANSFORMATIONS § A transformation (or function or mapping) T from to is a rule that assigns to each vector x in a vector T (x) in. § The set is called domain of T, and is called the codomain of T. § The notation indicates that the domain of T is and the codomain is. § For x in , the vector T (x) in is called the image of x (under the action of T ). § The set of all images T (x) is called the range of T. See the figure on the next slide. © 2012 Pearson Education, Inc. Slide 1. 8 - 5

MATRIX TRANSFORMATIONS § For each x in , T (x) is computed as Ax, where A is an matrix. § For simplicity, we denote such a matrix transformation by. § The domain of T is when A has n columns and the codomain of T is when each column of A has m entries. © 2012 Pearson Education, Inc. Slide 1. 8 - 6

MATRIX TRANSFORMATIONS § The range of T is the set of all linear combinations of the columns of A, because each image T (x) is of the form Ax. § Example 1: Let and define a transformation so that , , by , . © 2012 Pearson Education, Inc. Slide 1. 8 - 7

MATRIX TRANSFORMATIONS a. Find T (u), the image of u under the transformation T. b. Find an x in whose image under T is b. c. Is there more than one x whose image under T is b? d. Determine if c is in the range of the transformation T. © 2012 Pearson Education, Inc. Slide 1. 8 - 8

MATRIX TRANSFORMATIONS § Solution: a. Compute. b. Solve , or for x. That is,

MATRIX TRANSFORMATIONS § Row reduce the augmented matrix: ----(2) § Hence , , and

MATRIX TRANSFORMATIONS c. Any x whose image under T is b must satisfy equation (1). § From (2), it is clear that equation (1) has a unique solution. § So there is exactly one x whose image is b. d. The vector c is in the range of T if c is the image of some x in , that is, if for some x. § This is another way of asking if the system is consistent. © 2012 Pearson Education, Inc. Slide 1. 8 - 11

MATRIX TRANSFORMATIONS § To find the answer, row reduce the augmented matrix. § The third equation, system is inconsistent. , shows that the § So c is not in the range of T. © 2012 Pearson Education, Inc. Slide 1. 8 - 12

SHEAR TRANSFORMATION § Example 2: Let defined by transformation. . The transformation is called a shear § It can be shown that if T acts on each point in the square shown in the figure on the next slide, then the set of images forms the shaded parallelogram. © 2012 Pearson Education, Inc. Slide 1. 8 - 13

SHEAR TRANSFORMATION § The key idea is to show that T maps line segments onto line segments and then to check that the corners of the square map onto the vertices of the parallelogram. § For instance, the image of the point is , © 2012 Pearson Education, Inc. Slide 1. 8 - 14

LINEAR TRANSFORMATIONS and the image of § § is . T deforms the square as if the top of the square were pushed to the right while the base is held fixed. Definition: A transformation (or mapping) T is linear if: i. for all u, v in the domain of T; ii. for all scalars c and all u in the domain of T. © 2012 Pearson Education, Inc. Slide 1. 8 - 15

LINEAR TRANSFORMATIONS § Linear transformations preserve the operations of vector addition and scalar multiplication. If a transformation satisfies this relation for all u, v and c, d, it must be linear. © 2012 Pearson Education, Inc. Slide 1. 8 - 16

LINEAR TRANSFORMATIONS § Repeated application of (4) produces a useful generalization: ----(5) § In engineering and physics, (5) is referred to as a superposition principle. § Think of v 1, …, vp as signals that go into a system and T (v 1), …, T (vp) as the responses of that system to the signals. © 2012 Pearson Education, Inc. Slide 1. 8 - 17

LINEAR TRANSFORMATIONS § The system satisfies the superposition principle if whenever an input is expressed as a linear combination of such signals, the system’s response is the same linear combination of the responses to the individual signals. § Given a scalar r, define § T is called a contraction when dilation when. © 2012 Pearson Education, Inc. by . and a Slide 1. 8 - 18

Rotations § A linear transformation of the form is called a rotation by the

1 Matrix Algebra MATRIX OPERATIONS © 2012 Pearson Education, Inc.

MATRIX OPERATIONS § If A is an matrix—that is, a matrix with m rows and n columns—then the scalar entry in the ith row and jth column of A is denoted by aij and is called the (i, j)-entry of A. See the figure below. § Each column of A is a list of m real numbers, which identifies a vector in. © 2012 Pearson Education, Inc. Slide 2. 1 - 21

MATRIX OPERATIONS § The columns are denoted by a 1, …, and the matrix A is written as. § The number aij is the ith entry (from the top) of the jth column vector aj. § The diagonal entries in an matrix are a 11, a 22, a 33, …, and they form the main diagonal of A. § A diagonal matrix is a matrix whose nondiagonal entries are zero. § An example is the identity matrix, In. © 2012 Pearson Education, Inc. Slide 2. 1 - 22

SUMS AND SCALAR MULTIPLES § An matrix whose entries are all zero is a zero matrix and is written as 0. § The two matrices are equal if they have the same size (i. e. , the same number of rows and the same number of columns) and if their corresponding columns are equal, which amounts to saying that their corresponding entries are equal. § If A and B are matrices, then the sum is the matrix whose columns are the sums of the corresponding columns in A and B. © 2012 Pearson Education, Inc. Slide 2. 1 - 23

SUMS AND SCALAR MULTIPLES § Since vector addition of the columns is done entrywise, each entry in is the sum of the corresponding entries in A and B. § The sum same size. is defined only when A and B are the § Example 1: Let and © 2012 Pearson Education, Inc. . Find and . Slide 2. 1 - 24

SUMS AND SCALAR MULTIPLES § Solution: but is not defined because A and C have different sizes. § If r is a scalar and A is a matrix, then the scalar multiple r. A is the matrix whose columns are r times the corresponding columns in A. § Theorem 1: Let A, B, and C be matrices of the same size, and let r and s be scalars. § © 2012 Pearson Education, Inc. Slide 2. 1 - 25

SUMS AND SCALAR MULTIPLES b. c. d. e. f. § Each quantity in Theorem 1 is verified by showing that the matrix on the left side has the same size as the matrix on the right and that corresponding columns are equal. © 2012 Pearson Education, Inc. Slide 2. 1 - 26

MATRIX MULTIPLICATION § When a matrix B multiplies a vector x, it transforms x into the vector Bx. § If this vector is then multiplied in turn by a matrix A, the resulting vector is A (Bx). See the Fig. below. § Thus A (Bx) is produced from x by a composition of mappings—the linear transformations. © 2012 Pearson Education, Inc. Slide 2. 1 - 27

MATRIX MULTIPLICATION § Our goal is to represent this composite mapping as multiplication by a single matrix, denoted by AB, so that. See the figure below. § If A is , B is , and x is in , denote the columns of B by b 1, …, bp and the entries in x by x 1, …, xp. © 2012 Pearson Education, Inc. Slide 2. 1 - 28

MATRIX MULTIPLICATION § Then § By the linearity of multiplication by A, § The vector A (Bx) is a linear combination of the vectors Ab 1, …, Abp, using the entries in x as weights. § In matrix notation, this linear combination is written as. © 2012 Pearson Education, Inc. Slide 2. 1 - 29

MATRIX MULTIPLICATION § Thus multiplication by transforms x into A (Bx). § Definition: If A is an matrix, and if B is an matrix with columns b 1, …, bp, then the product AB is the matrix whose columns are Ab 1, …, Abp. § That is, § Multiplication of matrices corresponds to composition of linear transformations. © 2012 Pearson Education, Inc. Slide 2. 1 - 30

MATRIX MULTIPLICATION § Example 2: Compute AB, where and . § Solution: Write compute:

MATRIX MULTIPLICATION , , § Then Ab 1 © 2012 Pearson Education, Inc. Ab

MATRIX MULTIPLICATION § Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B. § Row—column rule for computing AB § If a product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding entries from row i of A and column j of B. § If (AB)ij denotes the (i, j)-entry in AB, and if A is an matrix, then. © 2012 Pearson Education, Inc. Slide 2. 1 - 33

PROPERTIES OF MATRIX MULTIPLICATION § Theorem 2: Let A be an matrix, and let B and C have sizes for which the indicated sums and products are defined. a. (associative law of multiplication) b. (left distributive law) c. (right distributive law) d. for any scalar r e. (identity for matrix multiplication) © 2012 Pearson Education, Inc. Slide 2. 1 - 34

PROPERTIES OF MATRIX MULTIPLICATION § The definition of AB makes x. However for all § AB does not always equal BA! § The position of the factors in the product AB is emphasized by saying that A is right-multiplied by B or that B is left-multiplied by A. © 2012 Pearson Education, Inc. Slide 2. 1 - 35

PROPERTIES OF MATRIX MULTIPLICATION § If , we say that A and B commute with one another. § Warnings: 1. In general, . 2. The cancellation laws do not hold for matrix multiplication. That is, if , then it is not true in general that. 3. If a product AB is the zero matrix, you cannot conclude in general that either or. © 2012 Pearson Education, Inc. Slide 2. 1 - 36

POWERS OF A MATRIX § If A is an matrix and if k is a positive integer, then Ak denotes the product of k copies of A: § If A is nonzero and if x is in , then Akx is the result of left-multiplying x by A repeatedly k times. § If , then A 0 x should be x itself. § Thus A 0 is interpreted as the identity matrix. © 2012 Pearson Education, Inc. Slide 2. 1 - 37

THE TRANSPOSE OF A MATRIX § Given an matrix A, the transpose of A is the matrix, denoted by AT, whose columns are formed from the corresponding rows of A. Theorem 3: Let A and B denote matrices whose sizes are appropriate for the following sums and products. § § § For any scalar r, § © 2012 Pearson Education, Inc. Slide 2. 1 - 38