Q25E If a subspace V of聽R3 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: Q25E (page 164)

If a subspace V of $R^{3}$ contains the standard vectors ${\overset{鈫�}{e}}_{1}, {\overset{鈫�}{e}}_{2}, {\overset{鈫�}{e}}_{3}$ then V must be $R^{3}$ .

Short Answer

Expert verified

The above statement is true.

If a subspace V of $R^{3}$ contains the standard vectors ${\overset{鈫�}{e}}_{1}, {\overset{鈫�}{e}}_{2}, {\overset{鈫�}{e}}_{3}$ then V must be $R^{3} .$

Step by step solution

Condition of a subspace to be equal to the vector space

Let V be a subspace of a vector space $R^{n} (F)$ , then the subspace V of $R^{n} (F)$ is equal to the vector space $R^{n} (F)$ if dim(V) = dim( $R^{n} (F)$ ) = n.

To show dim(V) = dim(R3)

Since the subspace V of $R^{3}$ contains the standard vectors ${\overset{鈫�}{e}}_{1}, {\overset{鈫�}{e}}_{2}, {\overset{鈫�}{e}}_{3}$ where

${\overset{鈫�}{e}}_{1} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], {\overset{鈫�}{e}}_{2} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], {\overset{鈫�}{e}}_{3} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]$

And the standard vectors ${\overset{鈫�}{e}}_{1}, {\overset{鈫�}{e}}_{2}, {\overset{鈫�}{e}}_{3}$ are linearly independent, so, these vectors form the bases of subspace V.

Thus, dimension of V = 3 = dimension of $R^{3} .$

Hence, V = $R^{3} .$

Final Answer

Since, dimension of V = 3 = dimension of $R^{3} .$

$鈬� V = R^{3} .$

Hence, if a subspace V of $R^{3}$ contains the standard vectors ${\overset{鈫�}{e}}_{1}, {\overset{鈫�}{e}}_{2}, {\overset{鈫�}{e}}_{3}$ then V must be $R^{3} .$

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91影视

If a subspace V of $R^{3}$ contains the standard vectors ${\overset{鈫�}{e}}_{1}, {\overset{鈫�}{e}}_{2}, {\overset{鈫�}{e}}_{3}$ then V must be $R^{3}$ .

Short Answer

Step by step solution

Condition of a subspace to be equal to the vector space

To show dim(V) = dim(R3)

Final Answer

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91影视

If a subspace V ofR3 contains the standard vectors e鈫�1,e鈫�2,e鈫�3then V must be R3.

Short Answer

Step by step solution

Condition of a subspace to be equal to the vector space

To show dim(V) = dim(R3)

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Calculus

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Mechanics Maths

Study anywhere. Anytime. Across all devices.

If a subspace V of $R^{3}$ contains the standard vectors ${\overset{鈫�}{e}}_{1}, {\overset{鈫�}{e}}_{2}, {\overset{鈫�}{e}}_{3}$ then V must be $R^{3}$ .