answer:First, we need to rewrite the equation of the line in slope-intercept form: frac{17 x}{10}-frac{9 y}{10}+frac{13}{10}=0 frac{9 y}{10}=frac{17 x}{10}-frac{13}{10} y=frac{17}{9}x-frac{13}{9} Now, we can use the point-slope form of a line to find the equation of the line perpendicular to the given line and passing through the given point: y-frac{18}{5}=-frac{9}{17}left(x+frac{11}{5}right) y=-frac{9}{17}x-frac{99}{17}+frac{18}{5} y=-frac{9}{17}x-frac{495}{85}+frac{306}{85} y=-frac{9}{17}x+frac{189}{85} Next, we can find the point of intersection of the two lines by solving the system of equations: frac{17}{9}x-frac{13}{9}=-frac{9}{17}x+frac{189}{85} frac{26}{9}x=frac{202}{85} x=frac{202}{85}cdotfrac{9}{26}=frac{182}{85} y=-frac{9}{17}left(frac{182}{85}right)+frac{189}{85} y=-frac{1638}{1445}+frac{189}{85} y=frac{189}{85}-frac{1638}{1445} y=frac{2593}{1445}-frac{1638}{1445} y=frac{955}{1445} Now, we can use the distance formula to find the distance between the given point and the point of intersection: d=sqrt{left(-frac{11}{5}-frac{182}{85}right)^2+left(frac{18}{5}-frac{955}{1445}right)^2} d=sqrt{left(-frac{11}{5}-frac{364}{170}right)^2+left(frac{18}{5}-frac{955}{1445}right)^2} d=sqrt{left(-frac{220}{170}-frac{364}{170}right)^2+left(frac{36}{10}-frac{955}{1445}right)^2} d=sqrt{left(-frac{584}{170}right)^2+left(frac{544}{1445}-frac{955}{1445}right)^2} d=sqrt{left(-frac{292}{85}right)^2+left(-frac{411}{1445}right)^2} d=sqrt{frac{85184}{7225}+frac{168481}{207025}} d=sqrt{frac{85184cdot3}{7225cdot3}+frac{168481}{207025}} d=sqrt{frac{255552}{21675}+frac{168481}{207025}} d=sqrt{frac{524033}{21675}} d=frac{sqrt{524033}}{sqrt{21675}} d=frac{sqrt{524033cdot185}}{sqrt{21675cdot185}} d=frac{142 sqrt{2}}{5sqrt{185}} Therefore, the distance from the point left(-frac{11}{5}, frac{18}{5}right) to the line frac{17 x}{10}-frac{9 y}{10}+frac{13}{10}=0 is frac{142 sqrt{2}}{5sqrt{185}}. The answer is frac{142 sqrt{2}}{5sqrt{185}}

answer:Let bar u be an arbitrary vector in V, which can be written as a unique linear combination of the basis vectors of V: bar u=sum_{i=1}^n u_ibar v_i Define the linear transformation T:V rightarrow W as: Tbar u = sum_{j=1}^m u_jbar w_j This transformation maps the basis vectors of V to the corresponding basis vectors of W, as required. It remains to show that T is linear. For any vectors bar u, bar v in V and scalar k, we have: T(bar u + bar v) = Tleft(sum_{i=1}^n u_ibar v_i + sum_{i=1}^n v_ibar v_iright) = sum_{j=1}^m (u_j + v_j)bar w_j = sum_{j=1}^m u_jbar w_j + sum_{j=1}^m v_jbar w_j = T(bar u) + T(bar v) T(kbar u) = Tleft(ksum_{i=1}^n u_ibar v_iright) = sum_{j=1}^m (ku_j)bar w_j = ksum_{j=1}^m u_jbar w_j = kT(bar u) Therefore, T is closed under addition and scalar multiplication, proving that it is a linear transformation.

answer:Let phi: G to text{Aut}(V) be a representation of G with character chi. Since Z(chi)=phi^{-1}(Z(phi(G))), then G/Z(chi)cong phi(G)/Z(phi(G)), hence phi(G)/Z(phi(G)) is abelian. Let gin Gsetminus Z(chi), then phi(g)inphi(G)setminus Z(phi(G)). Since phi(G)/Z(phi(G)) is abelian, then [phi(g),phi(h)]in Z(phi(G)) for all hin G. From Schur's lemma, we deduce that [phi(g),phi(h)]=lambda_h 1_V for some 0neqlambda_hinmathbb{C}. On the other hand, phi(g)inphi(G)setminus Z(phi(G)) follows that there exists hin G such that [phi(g),phi(h)]neq 1_V. Then phi(g)^{-1}phi(h)^{-1}phi(g)phi(h)=lambda_h 1_V, where lambda_hneq 1. Hence phi(h)^{-1}phi(g)phi(h)=lambda_hphi(g). Comparing traces of left and right-hand sides, we conclude that chi(g)=lambda_hchi(g) and chi(g)(1-lambda_h)=0. But lambda_hneq 1, so chi(g)=0.

answer:The broadening of spectral lines in a rotating star is caused by the Doppler effect, as light from different parts of the star moving towards or away from the observer is shifted in wavelength. This differs from pressure broadening, which is caused by collisions between atoms in the star's atmosphere, resulting in a wider range of energies for the emitted photons.