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Chapter 3: Q3.3-5E (page 143)
In Exercisesthrough, find the redundant column vectors of the given matrix 鈥渂y inspection.鈥� Then find a basis of the image ofand a basis of the kernel of .
The relation between the vectors is, . The basis of the image is,and the basis of the kernel is, .
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In Exercise 40 through 43, consider the problem of fitting a conic through given points in the plane; see Exercise 53 through 62 in section 1.2. Recall that a conic is a curve in that can be described by an equation of the form , where at least one of the coefficients is non zero.
43. How many conics can you fit through six distinct points? Describe all possible scenarios, and give an example in each case.
In Exercise 40 through 43, consider the problem of fitting a conic through given points in the plane; see Exercise 53 through 62 in section 1.2. Recall that a conic is a curve in that can be described by an equation of the form , where at least one of the coefficients is non zero.
41. How many conics can you fit through four distinct points?
For which value(s) of the constant k do the vectors below form a basis of ?
In Exercise 40 through 43, consider the problem of fitting a conic throughgiven pointsin the plane; see Exercise 53 through 62 in section 1.2. Recall that a conic is a curve inthat can be described by an equation of the form , where at least one of the coefficients is non zero.
40. Explain why fitting a conic through the points amounts to finding the kernel of anmatrix. Give the entries of the row of .
Note that a one-dimensional subspace of the kernel of defines a unique conic, since the equationsanddescribe the same conic.
Give an example of a linear transformation whose image is the line spanned by in .
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