1.2.15. (!) Let W be a closed walk of length at least 1 that does not contain a cycle. Prove that some edge of W repeats immediately (once in each direction).
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- Does either of P = (4, 11, 20) or Q = (-1, 6, 16) lie on the path r(t) = ( 1 + t, 2 + t 2, t 4)?Suppose that Mrs. Jena (in the position of T) want to travel to the top of regions diagonally right or left. It is impossible to stop on the STOP region and also not possible to jump too. Determine the possible number of routes that she is travelling to the top region. STOP TExercise 6
- ). 4. Rectangle DEFG with vertices D(-4, 3), E(O, 2), F(-2, -6), and G(-6, -5): (x, y) (x+ 4, y + 1) D'( E'( . F' G' - onent form. 7. OGina Wison (AI Things Algebra, LLC). 2015-20182. Suppose n ≥ 1 is an integer. Consider an (n + 1) x (n + 1) grid of integer points; i.e. points of the form (a, b) where 0 ≤ a,b ≤n. A Binomial Path with 2n steps is a path from the point (0, 0) to (n, n) formed by moving either 'right' (i.e. from (a, b) to (a +1, b)) or ‘up' (i.e. from (a, b) to (a, b+1)). (a) Draw all distinct Binomial Paths with 2n steps when n = = 2. (b) Write down a correspondence that relates the Binomial Paths with 2n steps to strings of length 2n consisting of exactly n 1s and n Os. More precisely, let B₁, be the set of Binomial Paths with 2n steps, and let Sn be the set of strings of length 2n consisting of exactly n 1s and n Os. Construct a bijection f: Bn Sn. (You don't have to prove that it is a bijection.)6. The covariance between X and Y is calculated and recorded as SXY (and sxy is a negative number). A linear transformation is performed on X. The linearly transformed variable is called W, where W; = 5 + Xi. The covariance between W and Y is calculated and recorded as sWY. Which statement is correct? A) В) In this example, swy would be less than sxY. In this example, swY Would be equal to sxY. In this example, swy would be greater than SXXY. 7. The covariance between X and Y is calculated and recorded as sxy (and sxy is a positive number). A linear transformation is performed on X. The linearly transformed variable is called W, where W; = (4)X;. The covariance between W and Y is calculated and recorded as sWY. Which statement is correct? In this example, sxY would be less than swy. In this example, sxY Would be equal to swy. In this example, sxY would be greater than sWY. A) B) 8. The correlation between X and Y is calculated and recorded as rxY (and rxy is positive). A linear…
- 18.8.2 Affine Independence 477 14. (T/F) Given S = {b1.....bx} in R", each p in aff S has a unique represcntation as an affine combination of bi...bk- %3D 15. (T/F) If S = {v.....v} is an affinely independent set in R" and if p in R" has a negative barycentric coordinate determincd by S, then p is not in aff S. 16. (T/F) When color information is specificd at cach vertex V1, V2. V3 of a triangle in R', then the color may be inter- polated at a point p in aff {v, V2, V3} using the barycentric coordinates of p. 17. (T/F) If VỊ, V2, V3. a, and b are in R3 and if a ray a +ib for 120 intersects the triangle with vertices v. V2, and v3, then the barycentric coordinates of the interscction point arc all nonnegative. pue 18. (T/F) If T is a triangle in R and if a point p is on an edge of the triangle, then the barycentric coordinates of p (for this triangle) arc not all positive8. Answer these two questions:
