Linear Algebra, Section 1.9

Linear Algebra, Section 1.9 First, some vocabulary: A function is a rule that associates objects in a set (the domain) to a unique object in a set (the codomain). The range or image of f is: {y|y = f (x)} We don’t talk about the codomain in calculus anymore for some reason... Think of the range (or image) as a subset of the codomain. In calculus, we have the following definitions. These are necessary before we can talk about the inverse of a function. • A function y = f (x) is said to be onto (its codomain) if, for every y (in the codomain), there is an x such that y = f (x). Note: Every function is automatically onto its image by definition (Since we only talk about the range in calculus, this is probably why the codomain is never mentioned anymore). Normally, the question is whether the function is onto its codomain. For example, y = x2 is not onto the real line, but is onto its range, which is the interval [0, ∞). If we don’t want to specify that a function is onto its codomain, we will say that f maps x into the codomain. • A function y = f (x) is said to be 1 − 1 if f (x1 ) = f (x2 ) ⇒ x1 = x2 or (this is logically equivalent to the equation above): x1 6= x2 ⇒ f (x1 ) 6= f (x2 ) In words, this means that each element of the range came from a unique element of the domain. In calculus, you might remember this graphically as the horizontal line test- If any horizontal line passes through the graph of f in more than one place, then f (x1 ) = f (x2 ), but x1 is not x2 . For example, y = x2 is not 1 − 1 because (−2)2 = 22 , but −2 6= 2. In Section 1.9, the big theorem was that: If T : IRn → IRm is a linear transformation, then there exists an m × n matrix A so that T (x) = Ax Additionally, A = [T (e1 ) T (e2 ) . . . T (en )] Another way to think of this is to say that every linear transformation from IRn to IRm can be realized as matrix-vector multiplication. Now we are looking at the equation Ax = b as a linear transformation, where x ∈ IRn is the domain, and IRm is the codomain. As it turns out, linear transformations are also continuous and differentiable, but to show this we will need to be able to measure distances in IRn and IRm . 1 1 Onto When will T (x) = Ax be onto? This would imply that for every b ∈ IRm , there is (at least one) solution to Ax = b. This is the setup for Theorem 4, page 43. We now list that theorem, together with our new terminology: Theorem 4, Revised: Let A be an m × n matrix. The following are logically equivalent: 1. The function Ax = b is onto IRm . 2. For each b ∈ IRm , the equation Ax = b has a solution. 3. Each b in IRm is a linear combination of the columns of A. 4. The columns of A span IRm . 5. A has a pivot position in every row. 2 One to One When will T (x) = Ax be 1-1? This means that the matrix equation Ax = b always has at most one solution (it might have no solution). Let’s write out the counterpart to Theorem 4. To neatly summarize, you should write out a comparison chart! The “One to One” Theorem: Let A be an m × n matrix. The following are logically equivalent: 1. The function Ax = b is 1 − 1. 2. The equation Ax = b has at most one solution for every b 3. The equation Ax = 0 has only the trivial solution. 4. The columns of A are linearly independent. 5. A has a pivot position in every column. 3 Worked Examples 1. Let T : IR2 → IR2 by mapping the span of (3, 2) and (−1, 1) to the span of (1, 1). Find a matrix that realizes this mapping, then determine if the mapping is 1-1 and/or onto: The mapping is not unique in this case because the question does not specify how the mapping performs this operation. However, since Ax is a linear combination of the columns of A, we know that the columns of A must be scalar multiples of (1, 1). Furthermore, we know that the span of 2 (3, 2) and (−1, 1), being linearly independent, is all of IR2 . We have lots of choices for A; one A might be:     2 1 2 1 A= ⇒ rref(A) = 2 1 0 0 This mapping is neither 1-1 nor onto, since there is only one pivot and two rows/columns. 2. Let A be m × n with m < n so that A is a “wide” matrix. If it has m pivots, then the corresponding transformation x → Ax will be onto IRm , but will not be 1 − 1 (because of the free variables). 3. Let A be m × n with m > n so that A is a “tall” matrix. If it has the maximum number of pivots possible, n, then the transformation x → Ax will be 1-1 (a pivot in every column, no free variables, so that Ax = 0 has only the trivial solution), but A will not be onto since there is not a pivot in every row. NOTE: The implication of numbers 2 and 3 is that, for x → Ax to be 1-1 and onto, the matrix must be square, and the RREF of A must be the identity matrix. 4. Construct a mapping x → Ax so that (−1, 1) is mapped to (3, 1) and (1, 1) is mapped to (2, 1). State whether the mapping is 1 − 1 and/or onto: We’ll have four unknowns for the entries of A. We cannot use our “big theorem” since we don’t know how the mapping transforms e1 and e2 , so we set up some equations:           a b −1 3 a b 1 2 = = c d 1 1 c d 1 1 Which translates to: −a + b a+b =3 =2 −c + d c+d =1 =1 Which translates to the following two augmented matrices:     −1 1 1 −1 1 3 1 1 2 1 1 1 Whose RREF are:     0 −1/2 1 0 0 1 5/2 0 1 1   −1/2 5/2 So the matrix A is , and the mapping x → Ax will be both 0 1 1 − 1 and onto. 1 0 Note: A linear mapping from IR2 to IR2 is uniquely defined by its action on two points. 3 4 Extra Practice with Functions We’ve introduced linear functions from IRn to IRm - how do we write nonlinear functions F from IRn to IRm ? First note that there are n inputs and m outputs to F . We will write an input as (x1 , . . . , xn ), and the outputs as (F1 , . . . , Fm ), where each Fi maps IRn to IR. Here’s an example before we make the general statement. Here, F maps IR2 to IR2 :  2    x1 + x22 F1 (x1 , x2 ) F (x) = = x2 − x1 F2 (x1 , x2 ) We write a generic function from IRn into IRm by its coordinate functions,   f1 (x1 , x2 , . . . , xn )  f2 (x1 , x2 , . . . , xn )    y = F (x) =   ..   . fm (x1 , x2 , . . . , xn ) where each fi : IRn → IR. Here are some examples: • F : IR2 → IR3 by:  x2 + sin(x) F (x, y) =  x(y − 2)  y 2 − 3xy  • F : IR3 → IR2 by:  F (x, y, z) = x2 − y 2 3xyz − 5  • Solving y = F (x) cannot generally be done in closed form. Normally, we would need to approximate a solution, then find a method to give us better and better approximations. Sometimes we get lucky, and the problem is easy enough to solve directly- For example, solve for x, y so that:  2    x − y2 0 = xy 1 so that x2 = y 2 and xy = 1. Therefore, (1, 1) and (−1, −1) are solutions. Why study linear functions? While general nonlinear functions may be the norm, many times we can replace the difficult, nonlinear function by its local, linear approximation (remember doing that with the tangent line?)... More on that later in class. 4