4.2 - Dr. Doom Lin Alg MATH1850/2050 Section 4.2 Page 1 of...

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Unformatted text preview: Dr. Doom Lin Alg MATH1850/2050 Section 4.2 Page 1 of 5 Chapter 4 - Euclidean Vector Spaces Section 4.2 - Linear Transformations from R” to Rm Functions from R" to Rm. Recall that a function is a rule, say f, together with two sets, say A and B, the domain and codomain of the function respectively. The rule associates with each element a of the domain, exactly one element b in the codomain; b is called the image of a under f. Two functions f1, f2 are equal, and we write f1 2 f2, if they have the same domain, and if f1(a) = f2(a) for all a in the domain. {may Cami») L Examples: f : R H R, = 3:2; gag): 3*": “(113)-— x-z {0.35% I we $A1 (A is THE Mo? 2. ouofl- {Nubian ) {Hz—am— 09. 100 cor)le A150 SA‘1 3. is m “TEE Pei—m at LE {‘ (ii) f 1R2 —> R3, f(% y) = ($2,:Uyw + y) I‘M» Kmmiu «(db—2) = (\,—z,—|) , (5—134) is 1% I-mxbi 0? 0:7.) UHDEL ‘— Definition When the function f has domain R" and codomain Rm, f is called a map or transformation from R" to Rm; we say that f maps R” into Rm, and we denote this by writing f : R" H Rm. In the special case when n = m, the transformation f : R” H R” is called an operator on R”. Examples: f : R4 H R3 defined by f(331,332,333,334) = (331 — 332 + 3347 331332 + 3337 331) 0" {(3% ax! )xsaxu) 3 (w! :‘01. 2‘35 ) “m6 “x: “‘Txrha' COL: X\X1+)¢3 (403: X\ Dr. Doom Lin Alg MATHl850/2050 Section 4.2 Page 2 of 5 Definition A transformation T : R" —> Rm whose defining equations are linear is called a linear transformation (or a linear operator, if m = That is, a linear transformation T : R" —> Rm is defined by equations of the form 101 = (“13171 -- @2332 -- . . . -- a1n$n w2 — a21$1 __ a22$2 __ - - - __ a2n$n wm = amlaj 1 __ am2$2 __ - - - __ amngcn or in matrix notation, w = Ax where A = [aw]. A is called the standard matrix for the linear transforma- tion T, and T is called multiplication by A. Examples: T : R3 —> R2 defined by 101 = 33171 + 23172 — $3 [wxzqux 102 = $1 — $2 + 23173 “'7' g a '5 Z ’l X [TV-[i -. T(*u¥z,7<sl= A 3‘3 L/‘v\—v‘ 9/ “""“‘”’ ital T (1|)xlfx5‘ 1 (x\—2'¥1+x5 3 x\"'¥1 ‘3‘; , x;*2X3> m: : 1% o \ 7- Some Notation Convention If T : R" —> Rm is multiplication by A, and if we need to emphasize the con- nection, we shall denote this by writing TA : R" —> Rm. That is, TAX = Ax. Finally, we write [T] to represent the standard matrix associated with the linear transformation T. Whenever notation is mixed, remember that [TA] 2 A. Dr. Doom Lin Alg MATH1850/2050 Section 4.2 Page 3 of 5 Example: Zero Transformation from R” to Rm Thar-Q “L ALL 2c=(x.,n.---,><«) k*O-xre'ctm. [.3 W.“ m" Om Example: Identity Operator on R" T3): 5 FoL ALL >_< row.“ [T1= L Example: Reflection Operators Sid T' [fit—a \é’ REFLECTk-Ofi \Jg 1HE ‘1—4046 \ TCxél = (-xwp S 76m = Dr. Doom Lin Alg MATH1850/2050 Section 4.2 Page 4 of 5 Example: Orthogonal Projection Operators T: \fl3a , T OCR-WW“. "Algal-01) 005 306’ PM“ Thaw—a.) = (x, M513] Example: Rotation Operators T -. N} am} uasiuw 31 mm» aoumcmocuwxss av NJ Anew: 6 T O‘Ha» m6 : [nae we Example: Dilation and Contraction Operators “flaw? aw? MK! La 17(39): 3525 [s A- minim) '3, o a t i o 3 o-X a a '5 T1: R39“); “Ilia 12“): i} is A Cadrme Dr. Doom Lin Alg MATHl850/2050 Section 4.2 Page 5 of 5 Definition If TA : R" —> R“ and TB : R“ —> Rm are linear transformations, then the composition of T3 with TA, denoted by TB 0 TA is defined by (TB 0 TA)(X) = TB(TA(X)) Note that this composition is linear, and [TB 0 TA] 2 BA and TB 0 TA 2 TBA. Examples: Say T1 : R2 —> R2 is rotation counterclockwise by 6 = 7r/2, say T2 : R2 —> R2 is projection on the y-axis, 16"“) ‘ 0“”) T: or: aux.) = T. Cum») [11‘ ME ‘4“1)? 7— o q] =T\ (09(1) GAE My); ‘ o = (301,0) T‘C“”“‘)‘ “#1) Well: i Z [Win] ink mat. Remark: Composition is not commutative Example: T2°T \ (34» #2) '~ T2 (T\(X~ 913) = 11C‘Ku XI) = (0),“) 7' T\ °T1(¥nxz) ...
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4.2 - Dr. Doom Lin Alg MATH1850/2050 Section 4.2 Page 1 of...

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