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Suppose $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is a basis for $\mathbb{R}^{3} .$ Show that $\operatorname{Span}\left\{\mathbf{v}_{2}-\mathbf{v}_{1}, \mathbf{v}_{3}-\mathbf{v}_{1}\right\}$ is a plane in $\mathbb{R}^{3} .[\text { Hint: What can }$you say about $\mathbf{u}$ and $\mathbf{v}$ when $\operatorname{Span}\{\mathbf{u}, \mathbf{v}\}$ is a plane? $]$

Therefore, the set $\left\{\mathbf{v}_{2}-\mathbf{v}_{1}, \mathbf{v}_{3}-\mathbf{v}_{1}\right\}$ is linearly independent.So, by the above result $\operatorname{Span}\left\{\mathbf{v}_{2}-\mathbf{v}_{1}, \mathbf{v}_{3}-\mathbf{v}_{1}\right\}$ is a plane in $\mathbb{R}^{3}$ .

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 1

Affine Combinations

Vectors

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from the hypotheses that the set containing V one v two V three services it can be proven that the set off factors 15 minutes free, one on 53 minutes 31 is a linearly independent set. This implies that the space generated by 15 minutes We won on 53 minutes. We won. Has I mentioned to on this is enough to conclude that this space is a plane. Now to prove that the said continue fee to minus V one and federal ministry one is linearly independent. Let's do it by contradiction So suppose there linearly independent that iss feta minus V one is a multiple off p three minutes We won so we can write the above equation as K minus one times fy one plus feet too minus K times fee three equal to zero on this is a linear combination. We're not all the coefficients or zero on this would imply that the V 1 15 23 are linearly independent. But since the set containing the missile bases in particular linearly independent and this is a contradiction and we prove what we wanted

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