Let the decision boundary be defined as . Consider the points and , which lie on the decision boundary. This gives us two equations:
Subtracting these two equations gives us .
Note that the vector lies on the decision boundary, and it is directed from to . Since the dot product is zero, must be orthogonal to , and in turn, to the decision boundary.
upport Vector Machines 1. Show mathematically that weight vector is orthogonal to the decision boundary. 2. Show that the distance from a point vector x on to a decision boundary line y(x)- +bish...
Thank you! Q1 Question 1 1 Point d(x, y) = (x – y| is a metric on R. O true O false Save Answer Q2 Question 2 1 Point The Euclidean distance formula d(x, y) = V(x1 – yı)2 + ... + (xn (21, ... , Xn) and y = (41, ..., Yn), is a metric on R”. - Yn)2, where x = O true O false Save Answer Q3 Question 3 1 Point Every metric on a vector space...
3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change when it is multiplied by an orthogonal matrix. 3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change...
Find the distance from the point with position vector y=[ 1,-3]| to the line through the origin parallel to y = [-2,4]. Give your answer rounded to 2 decimal places.
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be repeated more than twice. You should change the "names" of the dummy indices before taking the product of two summations...
5. Find parametric equations for the line through the point (0, 1,2) that is orthogonal to the line x = 1 + t, y 1-t, 2t, and intersects this line. (Hint: Try drawing this scenario in two dimensions, ie. draw two orthogonal lines and a point on each line away from the intersection. How would you find the direction vector?)
5. If ||2|| := VxTx is the usual (Euclidean) length of a vector x E R”, show that the vector Qx has the same length whenever Q is an orthogonal n xn matrix. If we define the angle between vectors x, y E R” as Z(x,y) := cos-1 -1 / xTy \ ||3||||y|| show that the angle between Qx and Qy is unchanged.
(1 point) The distance d of a point P to the line through points A and B is the length of the component of AP that is orthogonal to AB, as indicated in the diagram. = So the distance from P (-4,-5, -4) to the line through the points A = (1, -2, 3) and B = (-3, 2, -3) is | (1 point) The distance d of a point P to the line through points A and B is...
I will upvote! (2)()dz in the vector space Cº|0, 1] to find the orthogonal projection of f(a) – 332 – 1 onto the subspaco V (1 point) Use the inner product < 1.9 > spanned by g(x) - and h(x) - 1 proj) (1 point) Find the orthogonal projection of -1 -5 V = 9 -11 onto the subspace V of R4 spanned by -4 -2 -4 -5 X1 = and X2 == 1 -28 -4 0 -32276/5641 -2789775641 projv...
2. Find distance from point S(-2, 3, 4) to the line x = 3 - 2t, y = –2 + 3t, 2 = 5 - 6t Write plane equation passing through point S and par- allel to the given line. Show calculation steps clear and cleanly.
(1 point) The distance d of a point P to the line through points A and B is the length of the component of AP that is orthogonal to AB, as indicated in the diagram. So the distance from P = (0, 2) to the line through the points A = (-1,-1) and B=(3,0) is