Give a geometric description of Span{𝐯1, 𝐯2} for the vectors 𝐯1=[ 8 2 -6 ] and 𝐯2=[

step 1 Okay, so we're asked to give a geometric description of the span of V1 and V2, or V1 and V2 are

Give a geometric description of Span{𝐯1, 𝐯2} for the vectors...

Give a geometric description of $\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ for the vectors
$\mathbf{v}_{1}=\left[\begin{array}{r}{8} \\ {2} \\ {-6}\end{array}\right]$ and $\mathbf{v}_{2}=\left[\begin{array}{r}{12} \\ {3} \\ {-9}\end{array}\right]$

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Key Concepts

Span

The span of a set of vectors is the set of all linear combinations of those vectors. This means that any vector in the span can be written as a weighted sum of the given vectors, where the weights are real numbers.

Linear Combination

A linear combination is an expression made up of sums of scalar multiples of vectors. In forming the span, each vector is multiplied by a scalar and then all such products are added together, covering every possible vector in the subspace.

Linear Dependence

A set of vectors is linearly dependent if one of the vectors can be expressed as a scalar multiple of the other(s). In such a case, the additional vectors do not contribute new directions, and the span is lower-dimensional than it might otherwise be.

Geometric Interpretation

Geometrically, if the vectors are collinear (i.e., one is a scalar multiple of the other), their span is a line through the origin in the corresponding vector space. This line represents a one-dimensional subspace of the larger space.

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