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Q54E

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Found in: Page 37

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Consider two vectors ${\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{1}}}$ and ${\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{2}}}$ in R3 that are not parallel.Which vectors in localid="1668167992227" ${{\mathbf{ℝ}}}^{{\mathbf{3}}}$ are linear combinations of ${\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{1}}}$ and ${\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{2}}}$? Describe the set of these vectors geometrically. Include a sketch in your answer.

The vectors of the form ${c}_{1}{\stackrel{\to }{v}}_{1}+{c}_{2}{\stackrel{\to }{v}}_{2}$ a plane through the origin containing ${\stackrel{\to }{v}}_{1}$and ${\stackrel{\to }{v}}_{2}$.

See the step by step solution

## Step 1:

If vectors are not parallel then they are linearly independent. With the use parallelogram law of vector addition, we get that vector is linear combination of those 2 vectors if and only if it belongs to the same plane. Hence linear combination of the vectors ${\stackrel{\to }{v}}_{1}$ and ${\stackrel{\to }{v}}_{2}$ represents a plane i.e.

The vectors of the form ${c}_{1}{\stackrel{\to }{v}}_{1}+{c}_{2}{\stackrel{\to }{v}}_{2}$ a plane through the origin containing ${\stackrel{\to }{v}}_{1}$ and ${\stackrel{\to }{v}}_{2}$.

The vectors of the form ${c}_{1}{\stackrel{\to }{v}}_{1}+{c}_{2}{\stackrel{\to }{v}}_{2}$ a plane through the origin containing ${\stackrel{\to }{v}}_{1}$and ${\stackrel{\to }{v}}_{2}$ .