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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: Q49E (page 200)

if $f_{1}, f_{2}, f_{3}$ is a basis of linear space V and if f is any element of V then the elements $f + f_{1}, f + f_{2}, f + f_{3}$ must form a basis of V as well.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Determine the basis of the linear space

If $f_{1}, f_{2} . f_{3}$ is a basis of linear space V, and if f is any element of V, then the elements $f + f_{1}, f + f_{2}, f + f_{3}$ must form a basis of V as well.

02

Determine if the statement is True of false

Let $f = - f_{3}$ be an element of V.

Then the set as follows.

$f + f_{1}, f + f_{2} f + f_{3} = + f_{3} + f_{1} + f_{3} + f_{2}, f_{3} + f_{3} = - f_{3} + f_{1}, - f_{3} + f_{2}, 0$

The set contains two non-zero elements only instead of three elements.

Thus, $- f_{3} + f_{1}, - f_{3} + f_{2}, 0$ can鈥檛 be bases of V as the set.

Hence, $f + f_{1}, f + f_{2} + f + f_{3} {鈭�}_{i = 1}^{n} X_{i}^{2}$ can鈥檛 be bases of V.

Thus, the given statement 鈥淚f $f_{1}, f_{2}, f_{3}$ is a basis of linear space V, and if f is any element of V, then the elements $f + f_{1}, f + f_{2}, f + f_{3}$ must form a basis of V as well鈥� is false.

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Most popular questions from this chapter

In Exercise 72through 74, let $Z_{n}$ be the set of all polynomials of degree $鈮� n$ such that f(0) = 0.

73. Is the linear transformation $T (f (t)) = {鈭�}_{0}^{t} f (x) d x$ an isomorphism from $P_{n - 1}$ to $Z_{n}$ ?

State true or false, the space $R^{2 脳 3}$ is five-dimensional.

Find the image, kernel, rank, and nullity of the transformation $T$ in $T (f (t)) = [\begin{matrix} f (7) \\ f (11) \end{matrix}]$ from $P_{2}$ to $R$ .

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $T (f (t)) = {鈭�}_{- 2}^{3} f (t) d t$ from to $P_{2} t o 鈩�$ .

Show that if Tis a linear transformation from Vto W, then $T (0_{v}) = 0_{w}$ whererole="math" localid="1659425903549" $0_{v}$ and $0_{w}$ are the neutral elements of Vand W, respectively. If T is an isomorphism, show that $T^{- 1} (0_{w}) = 0_{v}$ .

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