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Math / Linear Algebra 4 / Chapter 1.4 / Problem 4

Linear Algebra | 4th Edition | ISBN: 9780130084514 | Authors: Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence

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For each list of polynomials in P3(K), determine whether the first polynomial can be

Chapter 1, Problem 4

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QUESTION:

For each list of polynomials in P3(K), determine whether the first polynomial can be expressed as a linear combination of the other two. (a) (b) (c) (d) (e) (f) x 3 - 3x 4- 5, x3 4- 2x2 - x + 1, x3 + 3x2 - 1 4x3 + 2x2 - 6, x3 - 2x2 + 4x + 1,3x3 - 6x2 + x + 4 -2x3 - 1 lx2 4- 3x 4- 2, x3 - 2x2 + 3x - 1,2x3 + x2 + 3x - 2 x 3 4- x2 + 2x 4-13,2x3 - 3x2 + 4x + 1, x3 - x2 + 2x 4- 3 x 3 - 8x2 + 4x, x3 - 2x2 + 3x - 1, x3 - 2x + 3 6x3 - 3x2 4- x + 2, x3 - x2 + 2x 4- 3,2x3 - 3x 4-

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QUESTION:

For each list of polynomials in P3(K), determine whether the first polynomial can be expressed as a linear combination of the other two. (a) (b) (c) (d) (e) (f) x 3 - 3x 4- 5, x3 4- 2x2 - x + 1, x3 + 3x2 - 1 4x3 + 2x2 - 6, x3 - 2x2 + 4x + 1,3x3 - 6x2 + x + 4 -2x3 - 1 lx2 4- 3x 4- 2, x3 - 2x2 + 3x - 1,2x3 + x2 + 3x - 2 x 3 4- x2 + 2x 4-13,2x3 - 3x2 + 4x + 1, x3 - x2 + 2x 4- 3 x 3 - 8x2 + 4x, x3 - 2x2 + 3x - 1, x3 - 2x + 3 6x3 - 3x2 4- x + 2, x3 - x2 + 2x 4- 3,2x3 - 3x 4-

ANSWER:

Step 1 of 6

Given polynomials are

Let us consider

Now,

Comparing the coefficients from both sides,

On solving, we get

Therefore,

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