mathLinAlgebra3-07-draft

# mathLinAlgebra3-07-draft - ARE211 Fall 2007 LECTURE#13 THU...

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Preliminary draft only: please check for Fnal version ARE211, Fall 2007 LECTURE #13: THU, OCT 11, 2007 PRINT DATE: AUGUST 21, 2007 (LINALGEBRA3) Contents 3. Linear Algebra (cont) 1 3.9. Linear Functions 1 3.10. The “graph” of a linear function from R 2 to R 2 3 3. Linear Algebra (cont) 3.9. Linear Functions A function with domain X and range Y is a rule that assigns a unique point in the range to every point in the domain. Notation: f : X Y . The image of f , denoted f ( X ), is the set of points in the range that are reached from some point in the domain, i.e., f ( X ) = { f ( x ) Y : x X } . Note: it’s not required that every point in the range of a function be reached from some point in the domain. This means that the image of a function is not the same as the range. Indeed, if Y is any superset of f ( X ), then we can write f : X Y . There is a lot of variation in language concerning the names that are assigned to f ( X ) vs Y . 1

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2 LECTURE #13: THU, OCT 11, 2007 PRINT DATE: AUGUST 21, 2007 (LINALGEBRA3) Some books refer to Y Some books even use the word “range” to refer to f ( X ). I follow the language used by the classical Analysis texts (e.g., Rudin) and will use the above terminology consistently. The graph of f : X Y is deFned as: graph f = { ( x ,y ) X × Y : y = f ( x ) } (1) The symbol × indicates the Cartesian product of the two sets. The result is a set of vectors made by pairing elements of the Frst set and elements of the second. ±ormally: X × Y = { ( x , y ) : x X, y Y } (2) A linear function is a function that satisFes additivity and proportionality, that is, f : X Y is a linear function if for all x , y X and all α R , f ( x + y ) = f ( x ) + f ( y ) Additivity f ( α x ) = αf ( x ) Proportionality Examples: any function whose graph is a straight line? Ans: no (In fact, these are properly called a ²ne functions). f ( x ) = 1 + x where x is a scalar. Ans: no. f ( x ) = ax where a and x are scalars. Ans: yes. f ( x ) = a · x where x and a are vectors. Ans: yes.
ARE211, Fall 2007 3 FACTS: Any linear function from R 1 to R 1 can be written in the form f ( x ) = ax , for some scalar a . Any linear function from R n to R 1 can be written in the form f ( x ) = a · x , for some n -vector a . Any linear function from R n to R m can be written in the form f ( x ) = A x , where A is a matrix with m rows and n columns. Note that the de±nition of a linear function doesn’t make any sense unless both the domain X and the image, I = f ( X ) are vector spaces. To see this note recall that we require for all x , y X and all α R , f ( x + y ) = f ( x ) + f ( y ) f ( α x ) = αf ( x ) but if X is not a vector space, then it is possible that x , y X but either x + y or α x isn’t in X , in which case f won’t be de±ned for these values. Similarly, if both f

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mathLinAlgebra3-07-draft - ARE211 Fall 2007 LECTURE#13 THU...

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