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

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

Key Concepts

Basis and Dimension

A basis for a subspace is a set of linearly independent vectors that spans the entire subspace. The number of vectors in the basis defines the dimension of that subspace. In contexts where two vectors span a plane, the subspace is considered two-dimensional.

Plane in ?³

In ?³, if two non-collinear vectors are taken, their span forms a plane through the origin. The plane is the geometric manifestation of all possible linear combinations of the two vectors, representing a two-dimensional subspace within the three-dimensional space.

Subspace

A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. The span of any set of vectors forms a subspace, which can be characterized by its dimension and is important in understanding the structure of vector spaces.

Linear Combination

A linear combination is an expression constructed from a set of terms by multiplying each term by a corresponding scalar and then summing the results. This concept is fundamental in determining how various vectors can combine to form new vectors within a given space.

Span

The span of a set of vectors is the collection of all possible linear combinations of those vectors. It essentially represents every vector that can be reached by scaling and adding these vectors together, and it forms a subspace of the overall vector space.

Transcript

00:01 The question reads, determine the span of the two vectors v1 and v2 for the given vectors in real three -dimensional space and describe it geometrically.

00:25 So the two vectors are 1 -1, 2, and 2 negative 1, 3, 3.

00:47 We could construct vectors of the form, some constant times v1 plus some constant times v2.

01:06 That general vector is going to be x, y, z, and it's going to be some constant times 1, 2, times 2, negative 1, 3.

01:32 Equating the x values, i get the available x values are 1 times c1 plus 2 times c2.

01:48 The available y values are the negative 1 times c1 minus c2.

02:00 And the available z values are 2 times c1 plus 3 times c2.

02:24 So i could write this in matrix form, even augmented matrix form, as 1 times c1 plus 2 times c2, negative 1 times c1 minus 1 times c2, 2 times c1, 3 times c2.

03:12 But i'm going to write this as an augmented matrix, x, y, z.

03:31 So now we can figure out what the values of c1 and c2 could be.

03:40 So i'm going to take the first row times negative 1 and add it to the second row.

03:48 No, no, no.

03:50 I'm just going to add it directly to the second row.

04:09 And now i'm going to multiply by negative 2 and add it to the third row.

04:24 Let me just double check my work, though.

04:29 Yeah, that's supposed to be x plus y.

04:33 Thought there was a mistake there.

04:35 All i was doing is adding the first two rows.

04:37 1 plus negative 1 is 0.

04:39 2 plus negative 1 is 1.