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1.4 Multiplication of Vectors and Linear Functions The previous three sections were concerned with vectors and vector operations. The next three sections look at multiplication of vectors and matrices and linear functions which are another important ingredient in applications of linear algebra. Multiplication and linear functions are closely related and we begin with the product of a row vector and a column vector. x1 x Definition 1. If a = (a1, a2, …, an) is a row vector and x = .2 is a column vector with xn the same number of components then the product ax of a and x is the sum of the products of corresponding components, i.e. (1) x1 n x ax = (a1, a2, …, an) .2 = a1x1 + a2x2 + … + anxn = aixi i=1 xn Example 1. 6 (2, -5, 4) -1 = (2)(6) + (-5)(-1) + (4)(0) = 12 + 5 + 0 = 17 0 The importance of this type of product is it gives us a compact way to express linear functions. Example 2. The linear function w = 2x – 4y + 7z can be written either as x w = (2, -4, 7) y = au z or 2 w = (x, y, z) -4 = vb 7 where 1.4 - 1 a = (2, -4, 7) x u = y z v = (x, y, z) 2 b = -4 7 Depending on the situation, it may be more natural to write the variables as a column vector and the coefficients as a row vector or vice versa. For example, often we make a set of prices a row vector and a set of corresponding amounts a column vector. Definition 2. A function z = (x1, x2, …, xn) of the variables x1, x2, …, xn is said to be linear if it has the form (2) x1 n x … z = (x1, x2, …, xn) = a1x1 + a2x2 + + anxn = ajxj = (a1, a2, …, an) .2 j=1 xn where a1, a2, …, an represent constants. In other words, a linear function of a certain collection of variables is one where the variables are multiplied by numbers and then added up or one that is equivalent to a function of this form. Putting it another way, if we think of the variables x1, x2, …, xn as x1 x a vector x = .2 , then a function z = (x) is linear if it has the form z = ax where xn a = (a1, a2, …, an). Any other function is called a non-linear function. Example 3. The following are linear functions y = 7x w = 2x – 4y + 7z y = 4x1 + 3x2 - 7x3 + 5x4 y = (x + 2)2 – x2 – 4 The last is linear because it is equivalent to y = 2x. The following are non-linear functions A = xy 1.4 - 2 p = 8.3NT V y = 4x1 + 3(x2)2 y = sin(x1) + 3x2 y = 7x + 2 w = 2x – 4y + 7z - 3 In many other mathematics courses the last two functions would be called linear functions. However, in this course they won’t. Instead they will be called inhomogeneous linear functions because they are a linear function plus a constant. Most functions are non-linear. However, linear functions occur so frequently in applications that they are worth a lot of attention. Furthermore, in many situations one can approximate a non-linear function by one or more linear ones. Example 4. An electronics company makes two types of circuit boards for computers, namely ethernet cards and sound cards. Each of these boards requires a certain number of resistors, capacitors and transistors as follows resistors capacitors transistors ethernet card 5 2 3 sound card 7 3 5 Let e s r c t = # of ethernet cards the company makes in a certain day = # of sound card the company makes in a certain day = # of resistors needed to produce the e ethernet cards and s sound cards = # of capacitors needed to produce the e ethernet cards and s sound cards = # of transistors needed to produce the e ethernet cards and s sound cards Then r = (5 resistors resistors ) (e ethernet cards) + (7 ) (s sound cards) ethernet card sound card or r = 5e + 7s So r is a linear function of e and s. Similarly, c = 2e + 3s t = 3e + 5s are linear functions. In this same context let 1.4 - 3 pr pc pr pe ps = = = = = price of a resistor price of a capacitor price of a transistor cost of all the resistors, inductors and transistors in an ethernet card cost of all the resistors, inductors and transistors in an sound card Then pe = 5pr + 2pe + 3pt ps = 7pr + 3pe + 2pt are again linear functions. Algebraic properties of the vector product. The vector product (1) has some of the familiar properties of regular products. x1 x Propostion 1. Let a = (a1, a2, …, an) and b = (b1, b2, …, bn) be row vectors and x = .2 xn y1 y and y = .2 be column vectors with the same number of components and let t be a yn number. Then the following hold. (3) a(x + y) = ax + ay (a distributive law) (4) (a + b)x = ax + bx (another distributive law) (5) (ta)x = t(ax) (an associative law) (6) a(tx) = t(ax) (another associative law) (7) ax = xTaT Proof. We prove (3) and leave the proof of the others for exercises. One has n n n n n i=1 i=1 i=1 i=1 i=1 a(x + y) = ai(x + y)i = ai(xi + yi) = [(aixi) + (ayi)] = aixi + aiyi = ax + ay. // An alternative characterization of linear functions. There is an alternative characterization of linear functions that allows one to extend the concept of linear functions to other vector spaces. 1.4 - 4 Propostion 2. Let a = (a1, a2, …, an) be a row vector with n components and let the function z = T(x) be defined by z = T(x) = ax = a1x1 + a2x2 + … + anxn for any column x1 x vector x = .2 with n components. Then the function T(x) has the following two xn properties. (8) T(x + y) = T(x) + T(y) (9) T(tx) = t T(x) for any vectors x and y and number t. Furthermore, any function z = T(x) satisfying (8) and (9) is a linear function in the sense of Definition 2. Proof. First suppose that z = T(x) is linear. To show that (8) holds note that T(x + y) = a(x + y) by definition. By Propostion 1 above one has a(x + y) = ax + ay. However, by definition, the right side is equal to T(x) + T(y), so (8) holds. To show that (9) holds note that T(tx) = a(tx) by definition. By Propostion 1 above one has a(tx) = t(ax). However, by definition, the right side is equal to tT(x), so (9) holds. Now suppose z = T(x) is a function satisfying (8) and (9). Let e = i 0 0 . 0 1 0 . 0 be the vector x1 x such that every component is zero except the ith. Note that x = .2 can be written as the xn linear combination x = x1e1 + x2e2 + … + xnen, so T(x) = T(x1e1 + x2e2 + … + xnen). Since z = T(x) satisfies (8) and (9) one has x1T(e1) + x2T(e2) + … + xnT(en). If we let ai = T(ei), then T(x) = a1x1 + a2x2 + … + anxn, so z = T(x) is a linear function in the sense of Definition 2. // The properties (8) and (9) of functions that are linear in the sense of Definition 2, are used to extend the concept of a linear function to general vectors. 1.4 - 5 Definition 3. Let z = T(x) be a function that assigns numbers or vectors z to all vectors x in some vector space. Then T is said to be linear if it satisfies (8) and (9), i.e. T(x + y) = T(x) + T(y) T(tx) = t T(x) Proposition 2 shows that Definitions 2 and 3 agrees when x is a column vector. Often books write Tx instead of T(x) if T is a linear function because T(x) acts like a product of T and x. Example 5. Let V be the vector space of all functions z = f(x) defined for 0 x 1 and let be a fixed point lying in the interval 0 1. If f is in V, let 1 Rf = f 2 Sf = f () 1 1 3 5 7 Tf = 4 [ f 8 + f 8 + f 8 + f 8 ] Then R, S and T are linear functions. We prove R is linear; the proofs that S and T are linear are similar. One has 1 1 1 R(f + g) = (f + g)2 = f2 + g2 = Rf + Rg, so R satisfies (8), Also, 1 1 R(tf) = (tf)2 = t[f2] = t(Rf), so R satisfies (9). // 1.4 - 6 Example 6. Let V be the vector space of all continuous functions z = f(x) defined for 0 x 1. If f is in V and 0 x 1, let 1 Rf = f(x) dx 0 1 Sf = f(x) sin(x) dx 0 x (Tf)(x) = f(t) dt 0 Then R and S are real valued linear functions. Furthermore and Tf is a continuous function defined for 0 x 1 and T is a linear function from V to V. We prove R is linear; the proofs that S and T are linear are similar. One has 1 1 1 1 R(f + g) = (f + g)(x) dx = (f(x) + g(x)) dx = f(x) dx + g(x) dx = Rf + Rg, so R satisfies (8), 0 0 1 0 1 0 1 Also, R(tf) = (tf)(x) dx = t(f(x)) dx = t f(x) dx = t(Rf), so R satisfies (9). // 0 0 0 Example 7. Let V be the vector space of all functions z = f(x) defined for 0 x 1 which df have the property that the derivative f (x) = dx is also defined for 0 x 1. Let W be the vector space of all functions z = f(x) defined for 0 x 1. If f is in V and 0 x 1, let df Tf = dx Then T is a linear function from V to W. d df dg To prove this note that T(f + g) = dx (f + g) = dx + dx = Tf + Tg, so T satisfies (8), Also, d 1 df T(tf) = dx (tf) = t(f(x)) dx = tdx, so T satisfies (9). // 0 1.4 - 7 Proposition 3. If V and W are vector spaces and S and T are linear functions from V to W and t is a real number, then S + T and tT are also linear functions from V to W. Thus the set of linear functions from V to W is a subspace of the vector space of all functions from V to W. Proof. We show S + T is linear; the proof that tT is linear is similar. One has (S + T)(x + y) = S(x + y) + T(x + y) = Sx + Sy + Tx + Ty = Sx + Tx + Sy + Ty = (S + T)x + (S + T)y, so S + T satisfies (8). Also, (S + T)(tx) = S(tx) + T(tx) = t(Sx) + t(Sy) = t(Sx + Tx) = t[(S + T)x], so S + T satisfies (9). // 1.4 - 8