Q12E If a subspace V of Rn聽contains ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q12E (page 164)

If a subspace V of $R^{n}$ contains none of the standard vectors $e_{1}, e_{2}, . . . e_{n}$ then V consists of the zero vector only.

Short Answer

Expert verified

The above statement is false.

If a subspace V of $R^{n}$ contains none of the standard vectors $e_{1}, e_{2}, . . . e_{n}$ then V need not consist of the zero vector only.

Step by step solution

Assuming a subspace V of Rn

Let us take n = 3.

Then, V be a subspace of $R^{3}$ such that

$V = \{(x, y, z) 鈭� R^{3} : x + y + z = 0\}$

Finding the vectors of V

Since, $V = \{(x, y, z) 鈭� R^{3} : x + y + z = 0\}$

$鈬� V = \{(x, y, - x - y) 鈭� R^{3} : z = - x - y\} 鈬� V = \{x (1, 0, - 1) + y (0, 1, - 1) : x, y 鈭� R^{3}\}$

$鈬� V = s p a n \{[\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}]\} 鈭� R^{3}$

Thus, the vectors of V are $[\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}] and [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}]$ and none of them is equal to standard vectors of $R^{3} .$

i . e . e_{1} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] {,e}_{2} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] {ande}_{3} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]

Final Answer

If, V is a subspace of $R^{3}$ such that

$V = \{(x, y, z) 鈭� R^{3} : x + y + z = 0\}$

Then, there exist vectors of V which are not zero vectors and not equal to standard vectors of .

Hence, the statement 鈥淚f a subspace V of $R^{n}$ contains none of the standard vectors $e_{1}, e_{2}, . . . e_{n}$ then V consists of the zero vector only鈥� is false.

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91影视

If a subspace V of $R^{n}$ contains none of the standard vectors $e_{1}, e_{2}, . . . e_{n}$ then V consists of the zero vector only.

Short Answer

Step by step solution

Assuming a subspace V of Rn

Finding the vectors of V

Final Answer

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91影视

If a subspace V of Rncontains none of the standard vectors e1,e2,...en then V consists of the zero vector only.

Short Answer

Step by step solution

Assuming a subspace V of Rn

Finding the vectors of V

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Pure Maths

Mechanics Maths

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.

If a subspace V of $R^{n}$ contains none of the standard vectors $e_{1}, e_{2}, . . . e_{n}$ then V consists of the zero vector only.