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Chapter 3: Q13E (page 131)
In Exercises10through 20 , use paper and pencil to identify the redundant vectors. Thus determine whether the given vectors are linearly independent.
13. .
The vectors are redundant and linearly dependent.
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Reflection T about the plane in .
In Exercises 21 through 25, find the reduced row-echelon form of the given matrix A. Then find a basis of the image of A and a basis of the kernel of A.
22.
A subspace of is called a hyperplane if is defined by a homogeneous linear equation
,
where at least one of the coefficients is nonzero. What is a dimension of a hyperplane in ? Justify your answer carefully. What is a hyperplane in ? What is it in ?
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 Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.
53..
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