During one term in Math 310, Min Lee took seven tests, the last of which carried twice the weight of each of the others when averages were computed. Min’s test-score vector for the term was [84,78,91,80,72,88,83]. Show that Min’s final average, a weighted average, can be calculated as a dot product of this vector with another seven-component vector. How can the teacher obtain a class list of test averages by multiplying two matrices?
multiply it by the vertical matrix
1
1
1
1
1
1
2
then you get a sum, divide by 8 and you have the weighted average.
ok thx
Well, let's calculate Min's final average, shall we?
First, we need to assign weights to the tests. Since the last test carries twice the weight of each of the others, we can assign the weights as follows: [1,1,1,1,1,1,2].
Now, let's calculate the dot product of the weights vector with Min's test-score vector:
(84*1) + (78*1) + (91*1) + (80*1) + (72*1) + (88*1) + (83*2)
Simplifying that expression, we get:
84 + 78 + 91 + 80 + 72 + 88 + 166
Taking the sum, we find that Min's final grade is:
659
So, Min's final average, as a weighted average, is 659.
Now onto the second part of your question, how can the teacher obtain a class list of test averages by multiplying two matrices?
To obtain a class list of test averages using matrix multiplication, the teacher can create two matrices:
Matrix A represents the test-score vectors of each student, with each row corresponding to a different student.
Matrix B represents the weight vector assigned to each test, with each column corresponding to a different test.
By performing matrix multiplication between Matrix A and Matrix B, the teacher can calculate the dot product of each student's test-score vector with the weight vector, giving the weighted average for each student.
The resulting matrix will have the same number of rows as Matrix A (number of students) and the same number of columns as Matrix B (number of tests). Each entry in the resulting matrix will represent the weighted average for a specific student and test.
So, through matrix multiplication, the teacher can obtain a class list of test averages.
To calculate Min's final average as a weighted average using the given test-score vector, we need to find another seven-component vector that represents the weights of each test.
Let's assume the weight vector is [w1, w2, w3, w4, w5, w6, w7], where w7 is twice the weight of the other tests.
To calculate the final average, we can take the dot product of the test-score vector [84, 78, 91, 80, 72, 88, 83] with the weight vector [w1, w2, w3, w4, w5, w6, w7].
The dot product is calculated by multiplying the corresponding elements and summing the results:
Final average = (84*w1) + (78*w2) + (91*w3) + (80*w4) + (72*w5) + (88*w6) + (83*w7)
This formula represents Min's final average as a weighted average of the test scores, where each test score is multiplied by its respective weight.
To obtain a class list of test averages by multiplying two matrices, the teacher would need to organize the test scores of all the students in a matrix and the corresponding weight values in another matrix.
Let's assume we have a matrix A that represents the test scores of all the students, where each row represents a student and each column represents a test. For example:
A = | 84 78 91 80 72 88 83 |
| 75 85 92 76 80 89 87 |
| ... |
Let's also assume we have a weight matrix W that represents the weights of each test, where each row represents a test and each column represents a student. For example:
W = | w1 w2 w3 w4 w5 w6 w7 |
| w1 w2 w3 w4 w5 w6 w7 |
| ... |
To obtain the class list of test averages, the teacher can multiply matrix A with matrix W. The resulting matrix, let's call it M, would have the same number of rows as matrix A (representing the number of students) and one column. Each entry in matrix M would represent the average test score for each student, calculated with the corresponding weight.
M = A * W
For example, if student 1 in matrix A has test scores [84, 78, 91, 80, 72, 88, 83], and the weight vector is [w1, w2, w3, w4, w5, w6, w7], multiplying them would give:
M[1,1] = (84*w1) + (78*w2) + (91*w3) + (80*w4) + (72*w5) + (88*w6) + (83*w7)
Repeat this process for each student in matrix A, and the resulting matrix M will contain the average test scores for each student in one column.