Consider the vectors defined in Figure E15.6. The set is the standard basis set. The set is an alternate basis set. The vector is a vector that we wish to represent with respect to the two basis...
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Consider the set of all complex numbers. This can be considered a vector space, because it satisfies the ten defining properties. We can also define an inner product for this vector space where is...
Consider the space of complex numbers. Let this be the vector space X, and let the basis for X be be the conjugation operator i. Find the matrix of the transformation relative to the basis set...
Consider the vector space of all continuous functions on the interval [0,1]. The set which is defined in the figure below, contains two vectors from this vector space.i. From these two vectors,...
Consider the set of polynomials of degree 1 or less. This is a linear vector space. One basis set for this space isUsing this basis set, the polynomial y = 2 + 4t can be represented asConsider the...
Consider the vector space of all piece wise continuous functions on the interval [0,1]. The set which is defined in Figure E15.2, contains two vectors from this vector space.Figure E15.2i. Generate...
Consider the vector space of all piecewise continuous functions on the interval [0, 1]. The set which is defined in Figure E15.1, contains three vectors from this vector space.i. Show that this set...
Using the following basis vectors, find an orthogonal set using Gram-Schmidt orthogonalization. (Check your answer using MATLAB.)
Recall the apple and orange pattern recognition problem of Chapter 3. Find the angles between each of the prototype patterns (orange and apple) and the test input pattern (oblong orange). Verify...
Which of the following sets of vectors are independent? Find the dimension of the vector space spanned by each set. (Verify your answers to parts (i) and (iv) using the MATLAB function rank.)
The next three questions refer to subsets of the set of real polynomials defined over the real line (e.g., 3+2t+6t2). Tell which of these subsets are vector spaces. If the subset is not a vector...
The three parts to this question refer to subsets of the set of real-valued continuous functions defined on the interval [0,1]. Tell which of these subsets are vector spaces. If the subset is not a...
A vector x can be expanded in terms of the basis vectors asThe vectors and can be expanded in terms of the basis vectors asi. Find the expansion for x in terms of the basis vectors ii. A vector...
Consider a perceptron network, with the following weights and bias.i. Write out the equation for the decision boundary.ii. Show that the decision boundary is a vector space. (Demonstrate that the 10...
What is the dimension of the vector space described in Problem P5.1?Problem P5.1Consider the single-neuron perceptron network shown in Figure P5.1. Recall from Chapter 3 (see Eq. (3.6)) that the...
Consider again the perceptron described in Problem P5.1. If , show that the decision boundary is not a vector space.Problem P5.1Consider the single-neuron perceptron network shown in Figure P5.1....
1. Two vectors from the vector space described in the previous problem (polynomials defined on the interval [-1, 1]) are 1 + t and 1 – t Find an orthogonal set of vectors based on these two...
1. Using the following basis vectors, find an orthogonal set using Gram-Schmidt orthogonalization.2. Consider the vector space of all polynomials defined on the inter val [-1, 1]. Show that is a...
Recall from Chapters 3 and 4 that one-layer perceptrons can only be used to recognize patterns that are linearly separable (can be separated by a linear boundary — see Figure 3.3). If two patterns...
1. Show that the set Y of nonnegative continuous functions is not a vector space.2. Which of the following sets of vectors are independent? Find the dimension of the vector space spanned by each...
Consider the single-neuron perceptron network shown in Figure P5.1. Recall from Chapter 3 (see Eq. (3.6)) that the decision boundary for this network is given by Wp + b =0. Show that the decision...
One variation of the perceptron learning rule iswhere is called the learning rate. Prove convergence of this algorithm. Does the proof require a limit on the learning rate? Explain.
1. Consider the set of all continuous functions that satisfy the condition ƒ(0) = 0. Show that this is a vector space.2. Show that the set of 2 × 2 matrices is a vector space.
Consider again the four-category classification problem described in Problems P4.3 and P4.5. Suppose that we change the input vector p3 toi. Is the problem still linearly separable? Demonstrate your...
The symmetric hard limit function is sometimes used in perceptron networks, instead of the hard limit function. Target values are then taken from the set [-1, 1] instead of [0, 1].i. Write a simple...
We want to train a perceptron network using the following training set:starting from the initial conditionsi. Sketch the initial decision boundary, and show the weight vector and the three training...
We want to train a perceptron network with the following training set:The initial weight matrix and bias arei. Plot the initial decision boundary, weight vector and input patterns. Which patterns...
We have two categories of vectors. Category I consists ofCategory II consists ofi. Design a single-neuron perceptron network to recognize these two categories of vectors.ii. Draw the network...
We have four categories of vectors.i. Design a two-neuron perceptron network (single layer) to recognize these four categories of vectors. Sketch the decision boundaries. ii. Draw the network...
Prove mathematically (not graphically) that the following problem is unsolvable for a two-input/single-neuron perceptron.
Solve the classification problem in Exercise E4.2 by applying the perceptron rule to the following initial parameters, and repeat parts (ii) and (iii) with the new solution. Exercise E4.2Consider...
Solve the classification problem in Exercise E4.2 by solving inequalities (as in Problem P4.2), and repeat parts (ii) and (iii) with the new solution. (The solution is more difficult than Problem...
The vectors in the ordered set defined below were obtained by measuring the weight and ear lengths of toy rabbits and bears in the Fuzzy Wuzzy Animal Factory. The target values indicate whether the...
Consider the classification problem defined below.i. Design a single-neuron perceptron to solve this problem. Design the network graphically, by choosing weight vectors that are orthogonal to the...
Consider again the four-class decision problem that we introduced in Problem P4.3. Train a perceptron network to solve this problem using the perceptron learning rule.Problem P4.3We have a...
Solve the following classification problem with the perceptron rule. Apply each input vector in order, for as many repetitions as it takes to ensure that the problem is solved. Draw a graph of the...
We have a classification problem with four classes of input vector. The four classes areDesign a perceptron network to solve this problem.
Convert the classification problem defined below into an equivalent problem definition consisting of inequalities constraining weight and bias values.
Solve the three simple classification problems shown in Figure P4.1 by drawing a decision boundary. Find weight and bias values that result in single-neuron perceptrons with the chosen decision...
We want to design a Hamming network to recognize the following prototype vectors:i. Find the weight matrices and bias vectors for the Hamming network.ii. Draw the network diagram. iii. Apply the...
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