Linear transformations• A map T : V → W between vector spaces is linear if◦ T (x + y) = T (x) + T (y)◦ T (cx) = cT (x)• Let A be an m × n matri x.T : Rn→ Rmdefined by T (x) = Ax is linear.• T : Pn→ Pn−1defined by T (p (t)) = p′(t) is linear.• The only lin e ar maps T : R → R are T (x ) = αx.Recal l thatT (0) = 0 for linear maps.• Linear maps T : R2→ R are of the form Txy= αx + βy.For i nstance,T (x, y) = xy is not linear: T2x2y2T (x, y)Example 1. Let V = R2and W = R3. Let T be the linear map such thatT11=104, T−11=1−20.• What is T04?04= 211+ 2−11T04= T211+ 2−11= 2T11+ 2T−11=208+2−40=4−48Let x1,, xnbe a basis for V .A linear mapT : V → W is determined by the values T (x1),, T (xn).Why?Take anyv in V .Writev = c1x1++ cnxn. (Possi ble, because {x1,, xn} spans V .)By linearity ofT ,T (v) = T (c1x1++ cnx) = c1T (x1) ++ cnT (xn).Armin [email protected] geometri c examplesWe consider som e linear mapsR2→ R2, whi c h are defined by matr ix mu ltiplication,that is, byxAx.In fact: all l inear mapsRn→ Rmare given by xAx, for some matrix A.Example 2.The matrix A =c 00 cgives the map xcx, i.e.stretches every vector in R2by the s a me factor c.Example 3.The matrix A =0 11 0gives the mapxyyx, i.e.reflects every vector in R2through the line y = x.Example 4.The matrix A =1 00 0gives the mapxyx0, i.e.projects every vector in R2through onto the x-axis.Armin [email protected] 5.The matrix A =0 −11 0gives the mapxy−yx, i.e.rotates every vector in R2counter-clockwise by90◦.Representing linear maps by matricesDefinition 6. (From linear maps to matrices)Let x1,, xnbe a basis for V , and y1,, yma basis for W .The matrix representing T with respect to these bases• has n colum ns (one for each of the xj),• the j-th column has m entries a1,j,, am, jdetermined byT (xj) = a1,jy1++ am,jym.Example 7.Recall the map T given byxyyx.(reflects every vector in R2through the line y = x)• Which matrix A r e presents T with respect to thestandard bases?• Which matrix B represents T w ith respect to thebasis11,−11?Solution.•T10=01. Hence, A =0 ∗1 ∗.T01=10. Hence, A =0 11 0.If a linear map T : Rn→ Rmis represented by the matrix A with respect to t h estandard bases, then T (x) = Ax.Armin [email protected] multiplication corresponds t o function composition!That is, ifT1, T2are represent ed by A1, A2, then T1(T2(x)) = (A1A2)x.• T11=11= 111+ 0−11. Hence, B =1 ∗0 ∗.T−11=1−1= 011−−11. Hence, B =1 00 −1.Example 8. Let T : R2→ R3be the linear map such thatT10=123, T01=407.What is the ma t r ix B representing T with respect to the f ollow ing bases?11x1,−12x2for R2,111y1,010y2,001y3for R3.Solution. This time:T (x1) = T11= T10+ T01=123+407=5210= 5111− 3010+ 5001can you see it?otherw ise: do it!B =5 ∗−3 ∗5 ∗T (x2) = T−12= −T10+ 2T01= −123+ 2407=7−211= 7111− 9010+ 4001B =5 7−3 −95 4Tedious, even in this simple example! (But we can certainly do it.)A matrix representing T enc odes in colu mn j the coefficients of T (xj) expressed asa linear combination of y1,, ym.Armin [email protected] problemsExample 9. Let T : R2→ R2be the map wh ich rotates a vector counter-clockwise byangle θ.• Which matrix A represents T with respec t to the standard bases?• Verify that T (x) = Ax.Solution. Only keep reading if you need a h int!The first basis vec tor10gets send tocosθsin θ.Hence, the first column ofA isArmin
View Full Document
Unlocking...