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Linear Algebra With Applications
Found in: Page 145
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

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. (0,0),(1,0),(2,0),(0,1),(1,1),(2,1),(0,2),(1,2),(3,2).

Thus, the cubic that passes through the nine given points is of the form 2c10y-3c10y2+c10y3=0.

See the step by step solution

Step by Step Solution

Step 1: Given in the question.

Each point Pixi,yi defines an equation in the 10 variables c1,c2,.....,c10 given by:

role="math" localid="1660365952521" c1+Xic2 +yic3 +Xi2c4 +Xiyic5 +yi3c6 +yi3c7+xi2yic8 +xiyi2c9 +yi3c10=0,

There are nine points.

The system of nine equations is written as follows:

Ac=0Here, A=1x1y1x12x1y1y12x13x12y1x1y12y131x2y2x22x2y2y22x23x22y2x2y22y231x3y3x32x3y3y32x33x32y3x3y32y331x9y9x92x9y9y92x93x92y9x9y92y93

Step 2: Apply gauss-Jordan elimination in the matrix A.

Plug in the nine points to derive the A matrix.


Now, use gauss-Jordan elimination to solve the system Ac=0. Note that the A matrix is identical to the A matrix from Exercise 52, with the addition of one row. Thus, the first eight rows is replaced with row echelon form in Exercise 52.


Step 3: Showing that cubics through (0,0),(1,0),(2,0),(0,1),(1,1),(2,1),(0,2),(1,2),(3,2).

The solution of the equation Ac=0 which satisfies:


While c10 are free variables. Recall that the cubic equation is as follows:

c1+Xc2 +yc3 +X 2c4 +Xyc5 +y 2c6 +x3c7 +x2yc8 +xy2c9+y3c10=0

Therefore, the cubic that passes through the nine given points is of the form


Step 3: Sketch of cubics.

As the first example, substitutec7=1,c10=0. The cubic is


As the first example, substitutec7=0,c10=1. The cubic isyy-1y-2=0.

Now, for a pointx,yon the cubic curve is eithery=0,y=1 or y=2. This set is graphed as follows:

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