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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q41E (page 200)

TRUE OR FALSE?
41. If $f_{1} . . . . . . . f_{n}$ are polynomials such that the degree of $f_{k}$ is k (for k=1鈥� n), thenrole="math" localid="1659427620950" $f_{1} . . . . . . . f_{n}$ must be linearly independent.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

The linearity of a polynomial

Let $f_{i} = a_{i} x^{i} + . . .,$ i.e., the leading term of $f_{i}$ is $a_{i} x^{i} .$ Since $f_{i}$ is of degree i .

So, $a_{i} 鈮� 0$ for all i=1, 2, 3鈥� n.

Consider the equation

$c_{1} f_{1} + c_{2} f_{2} + . . . . . + c_{n} f_{n} = 0$

Equate the term of degree n in both sides of the above equation, then $c_{n} a_{n} = 0,$ , which implies, $c_{n} = 0$ as. $a_{n} 鈮� 0$ Then the above equation becomes $c_{1} f_{1} + c_{2} f_{2} + . . . . . + c_{n - 1} f_{n - 1} = 0 .$ .

Now equate the term of degree n-1 in both sides of the above equation, then it obtains $c_{n - 1} a_{n - 1} = 0$ , which implies, $c_{n - 1} = 0, a s a_{n - 1} 鈮� 0 .$

Proceeding in this way, we can conclude that $c_{i} = 0$ for all i=1, 2, 3,鈥.

Hence, the statement is true.

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Most popular questions from this chapter

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $T (f (t)) = f (7)$ from $P_{2}$ torole="math" localid="1659412169328" $鈩�$ .

TRUE OR FALSE

6. if $f_{1}, . ., f_{n}$ is a basis of a linear space V, then any element of V can be written as a linear combination of $f_{1}, . . ., f_{n}$ .

In Exercises 5 through 40, find the matrix of the given linear transformationwith respect to the given basis. If no basis is specified, use standard basis: $渭 = (1, t, t)$ for $P_{2}$ ,

role="math" localid="1659423247247" $渭 = ([\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}])$

for $^{2 脳 2}$ androle="math" localid="1659421462939" $渭 = (1, i)$ for,.For the space $U^{2 脳 2}$ of upper triangular $2 脳 2$ matrices, use the basis

$渭 = ([\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}])$

Unless another basis is given. In each case, determine whether Tis an isomorphism. If Tisn鈥檛 an isomorphism, find bases of the kernel and image of Tand thus determine the rank of T.

12. T $(M) = M [\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}]$ from $R^{2 脳 2}$ to $R^{2 脳 2}$ .

State true or false, the linear transformation T (f) = f (4t - 3 ) from P toP is an isomorphism.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $T (M) = M [\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}] - [\begin{matrix} 3 & 0 \\ 0 & 4 \end{matrix}] M$ from ${鈩�}^{2 脳 2} {迟辞鈩�}^{2 脳 2}$ to .

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