Determine span{𝐯1, 𝐯2} for the given vectors in ℝ^3, and describe it geometrically. 𝐯1=(1,2,-1),

Determine span{𝐯1, 𝐯2} for the given vectors in ℝ^3, and...

Determine $\operatorname{span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ for the given vectors in $\mathbb{R}^{3},$ and describe it geometrically.
$$\mathbf{v}_{1}=(1,2,-1), \mathbf{v}_{2}=(-2,-4,2)$$

Key Concepts

Span of a Set of Vectors

The span of a set of vectors is the collection of all possible linear combinations of those vectors. It defines the smallest subspace that contains the original vectors and is a fundamental concept in linear algebra for describing the space that can be reached using the given directions.

Linear Dependence

Linear dependence occurs when one vector in a set can be expressed as a scalar multiple or a combination of other vectors in the set. This concept helps in determining the dimension of the span; if vectors are linearly dependent, they do not add new directions beyond what is already spanned by the others.

Vector Subspaces

A vector subspace is a subset of a vector space that is itself a vector space, being closed under addition and scalar multiplication. In R³, subspaces can be a line, a plane, or the entire space, depending on the number of independent directions spanned by the given vectors.

Geometric Interpretation of the Span

The geometric interpretation of the span involves understanding how the linear combinations of vectors form geometrical objects. For example, if vectors are linearly dependent, like when one is a scalar multiple of the other, their span is just a line through the origin; if they were independent, their span might be a plane. This visual approach aids in conceptualizing the abstract algebraic operations.

Transcript

00:01 Determine the span of v1 and v2 for real three -dimensional space and describe it geometrically when v1 is 1 -2 -negative 1, and v2 is negative 1.

00:20 And v2 is negative 2, negative 4.

00:28 All right.

00:30 Well, all of the vectors are going to be of the form.

00:36 Form c1 v1 plus c2 v2.

00:49 X, y, z will be of the form c1 times 12 negative 1 plus c2 times negative 2, negative 4, 2.

01:10 All right.

01:15 So we know that the x value will be c1 minus 2c2.

01:25 We know that the y value will be 2c1 minus 4c2.

01:34 And we know that the z value will be negative c1 plus 2c2.

01:44 And now we can write our span as the span of, whoops, of the vectors v1 and v2 is all vectors in three -dimensional real space, for which the vector is as an x value, which is going to be c1 minus c1, minus 2c2.

02:40 It's going to have a y value that is 2c1 minus 4c2, and it's going to have a z value of negative c1 plus 2c2, where c1 and c2 are real numbers.

03:24 But what does this look like? the way i choose to decide what this looks like is that i will make a matrix comprised of 1, negative 2, 2, negative 4, negative 1, 2, augmented x, y, z.

03:57 So those numbers came from 1, negative 2, to negative 4, negative 1, 2.

04:12 Use some elementary row operations here.

04:16 We're going to multiply the first row times negative 2 and add it to the second row.

04:32 Um, negative 2 times negative 2 is positive 4.

04:43 And we add that to the second row and we get 0.

04:50 Negative 2 times negative 2 is 4 added the second row.

04:57 We get 0.

04:58 And so that's going to be negative 2x plus y...