Find the orthogonal decomposition of v with respect to W. 1 --[-]-~-~~-~(:][-D) -2 W = span 2 1 V projw(v) = perpw(v) = 4 3 000 000 1 -1 1

Find the cross product x where = 71-7j -4k and 7 = (-4,-3, 4). úxv=

Compute the orthogonal projection of u onto v. Use the square root symbol '√' where needed to give an exact value for your answer. 3 ▪- [21] V-8 u= = 1 0 -[8] proj,u =

Find the bilinear transformation which maps the points (-1,0,1) into points (0,i,3i)

Find a linear mapping G that maps [0, 1] x [0, 1] to the parallelogram in the xy-plane spanned by the vectors (-3,7) and (9,3). (Use symbolic notation and fractions where needed. Give your answer in the form (*, *).) G(u, v) =

Find the scalar equation of the line through (3,-2) and perpendicular to the line r = (4,-3) + t(-5,1) .  Please answer.  Thank you

3 = [ ¹₂ ], b = [ ³ ], c = [ ¹₁ ], d = [¯^¹'], and v = [¯2²], 5. Given vectors a = (a) Find all real numbers x₁, x2, x3, x4, such that x₁a + x₂b + x3c + x4d = v (b) Write [at least] one sentence about what you have done above using the following mathematical term: linear combination. (c) Write [at least] one sentence about what you have done above using the following mathematical term: span.

Find a linear map 7 that maps [0, 1] × [0, 1] to the parallelogram P in the xy-plane spanned by the vectors (6,5) and (7,4). (Give your answer in the form (*, *). Express numbers in exact form. Use symbolic notation and fractions where needed.) T(u, v) = (6u+7v,5u +4v) Calculate the double integral of e2x-over P via change of variables. (Express numbers in exact form. Use symbolic notation and fractions where needed.) 1/₂ 2²x-A= Incorrect 10 ½ (e¹⁰ − 1)(e² - 1) 70 CA

Let Ly be the line passing through the point P-(10, 1, -10) with direction vector d-[-2, -2, 3]", and let L2 be the line passing through the point P-(-10, 21, -6) with direction vector d,-[-2, -6, -1]". Find the shortest distance d between these two lines, and find a point Q, on L, and a point Q, on L2 so that d(Q7 Q2) = d. Use the square root symbol W where needed to give an exact value for your answer. d = 0 Qi = 1%3D(0,0,0) Q2 = (0, 0, 0)

If vector A = [4,1] and vector B = [-3,2] and A-B = [-1,-1]  Sketch the computation above and represent it as a vector addition