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Suppose X is a random vector, where X = (X(1), . . . , x(d))T , d with mean 0 and covariance matr...
Suppose X is a random vector, where X = (X(1), . . . , x(d))T , d with mean 0 and covariance matrix vv1 , for some vector v ER 1point possible (graded) Let v = . (i.e., v is the normalized version of v). What is the variance of v X? (If applicable, enter trans(v) for the transpose v of v, and normv) for the norm |vll of a vector v.) Var (V STANDARD NOTATION SubmitYou have used 0...
Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in...
Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in class is a valid norm on R2 (b) Prove that in R2, any open ball in 12 ("Euclidean metric") can be enclosed in an open ball in the loo norm ("sup" norm). (c). Say I have a collection of functions f:I R. Say I (1,2). Consider the convergence of a sequence of functions fn (z) → f(x) in 12-Show that the convergence amounts to...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
8. More generally, let X be any infinite-dimensional vector space equipped with an inner product ,)...
8. More generally, let X be any infinite-dimensional vector space equipped with an inner product ,) in such a way that the induced metric is complete. In particular, there is a norm on X defined by and the metric is given by d(r, y) yl Let A denote the unit ball A x E X < 1} We know that A is closed and bounded essentially from the definitions. Show that A is not compact. (Hint: Construct a sequence xn...
Problem 1. Let (X, d) be a metric space and t the metric topology on X....
Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.
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...
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)- +bishere | lwl| is the Euclidean norm. y(xq) llwll
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)- +bishere | lwl| is the Euclidean norm....
Question 1. Give an example of a complete metric space (X, d) and a function f...
Question 1. Give an example of a complete metric space (X, d) and a function f :X + X such that d(f(x), f(y)) < d(x, y) for all x, y E X with x + y and yet f does not have a fixed point. a map f:X + X has a fixed point if there is an element a E X such that f(a) = a.
is some norm on a vector spad then there is some inner product such that 1||...
is some norm on a vector spad then there is some inner product such that 1|| ||is some norm on a vector space V, then there is some inner product (:, - ) such that ||0||2 = (v, v) for all v E V. Select one: O True False
Consider d on R defined by d(x, y) = ?|x − y|. (1) Show that (R,...
Consider d on R defined by
d(x, y) = ?|x − y|.
(1) Show that (R, d) is a metric space.
(2) Show that the path γ(t) = t, t ∈ [0, 1] has infinite
length.
Remark: On (2), you only need to verify by the partitions of
equal distances. Although this is slightly different from the
actual definition, it indeed implies that length equals to
infinity, by using some techniques in the Riemann sum (e.g.
refining a partition). This...
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) =...
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) = / If(x)-g(x)| dx. 9) Let (X, di) and (Y, d2) be metric spaces. a) Prove that X × Y is a metric space with the metric b) Prove that X x Y is a metric space with the metric
Suppose X is a random vector, where X = (X(1), . . . , x(d))T , d with mean 0 and covariance matrix vv1 , for some vector v ER 1point possible (graded) Let v = . (i.e., v is the normalized version of v). What is the variance of v X? (If applicable, enter trans(v) for the transpose v of v, and normv) for the norm |vll of a vector v.) Var (V STANDARD NOTATION SubmitYou have used 0...
Part 2: Metrics and Norms 1. Norms and convergence: (a) Prove the l2 metric defined in class is a valid norm on R2 (b) Prove that in R2, any open ball in 12 ("Euclidean metric") can be enclosed in an open ball in the loo norm ("sup" norm). (c). Say I have a collection of functions f:I R. Say I (1,2). Consider the convergence of a sequence of functions fn (z) → f(x) in 12-Show that the convergence amounts to...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
8. More generally, let X be any infinite-dimensional vector space equipped with an inner product ,) in such a way that the induced metric is complete. In particular, there is a norm on X defined by and the metric is given by d(r, y) yl Let A denote the unit ball A x E X < 1} We know that A is closed and bounded essentially from the definitions. Show that A is not compact. (Hint: Construct a sequence xn...
Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.
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)- +bishere | lwl| is the Euclidean norm. y(xq) llwll
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)- +bishere | lwl| is the Euclidean norm....
Question 1. Give an example of a complete metric space (X, d) and a function f :X + X such that d(f(x), f(y)) < d(x, y) for all x, y E X with x + y and yet f does not have a fixed point. a map f:X + X has a fixed point if there is an element a E X such that f(a) = a.
is some norm on a vector spad then there is some inner product such that 1|| ||is some norm on a vector space V, then there is some inner product (:, - ) such that ||0||2 = (v, v) for all v E V. Select one: O True False
Consider d on R defined by
d(x, y) = ?|x − y|.
(1) Show that (R, d) is a metric space.
(2) Show that the path γ(t) = t, t ∈ [0, 1] has infinite
length.
Remark: On (2), you only need to verify by the partitions of
equal distances. Although this is slightly different from the
actual definition, it indeed implies that length equals to
infinity, by using some techniques in the Riemann sum (e.g.
refining a partition). This...
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) = / If(x)-g(x)| dx. 9) Let (X, di) and (Y, d2) be metric spaces. a) Prove that X × Y is a metric space with the metric b) Prove that X x Y is a metric space with the metric
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