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Add one vector to the set S = so that S is a basis for R

Add one vector to the set S = so that S is a basis for R

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The task is to add one vector to the set \( S = \left\{ \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \right\} \) so that \( S \) becomes a basis for \( \mathbb{R}^3 \). Currently, the set \( S \) contains two vectors. To form a basis for \( \mathbb{R}^3 \), which is the three-dimensional Euclidean space, a third linearly independent vector needs to be added to the set. This new vector, when combined with the existing ones, must span \( \mathbb{R}^3 \) and maintain linear independence among all three vectors.
Transcribed Image Text:The task is to add one vector to the set \( S = \left\{ \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \right\} \) so that \( S \) becomes a basis for \( \mathbb{R}^3 \). Currently, the set \( S \) contains two vectors. To form a basis for \( \mathbb{R}^3 \), which is the three-dimensional Euclidean space, a third linearly independent vector needs to be added to the set. This new vector, when combined with the existing ones, must span \( \mathbb{R}^3 \) and maintain linear independence among all three vectors.
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