In Exercises 15-18, find a basis for the space spanned by...

In Exercises $15-18,$ find a basis for the space spanned by the given vectors, $\mathbf{v}_{1}, \ldots, \mathbf{v}_{5}$ $$ \left[\begin{array}{r}{1} \\ {0} \\ {-3} \\ {2}\end{array}\right],\left[\begin{array}{r}{0} \\ {1} \\ {2} \\ {-3}\end{array}\right],\left[\begin{array}{r}{-3} \\ {-4} \\ {1} \\ {6}\end{array}\right],\left[\begin{array}{r}{1} \\ {-3} \\ {-8} \\ {7}\end{array}\right],\left[\begin{array}{r}{2} \\ {1} \\ {-6} \\ {9}\end{array}\right] $$

In Exercises $15-18,$ find a basis for the space spanned by the given vectors, $\mathbf{v}_{1}, \ldots, \mathbf{v}_{5}$
$$
\left[\begin{array}{r}{1} \\ {0} \\ {-3} \\ {2}\end{array}\right],\left[\begin{array}{r}{0} \\ {1} \\ {2} \\ {-3}\end{array}\right],\left[\begin{array}{r}{-3} \\ {-4} \\ {1} \\ {6}\end{array}\right],\left[\begin{array}{r}{1} \\ {-3} \\ {-8} \\ {7}\end{array}\right],\left[\begin{array}{r}{2} \\ {1} \\ {-6} \\ {9}\end{array}\right]
$$

Key Concepts

Basis

A basis is a set of vectors that is both linearly independent and spans the vector space. It provides a minimal and unique way of representing every vector in the space as a linear combination of the basis vectors.

Span

The span of a set of vectors is the collection of all possible linear combinations of those vectors. It defines a subspace and shows how a given set of vectors can generate all elements of that subspace.

Linear Independence

A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others. This property is essential for ensuring that the vectors in a basis provide a unique representation for each element of the space.

Row Reduction

Row reduction, often performed using Gaussian elimination, is a systematic method for simplifying matrices to row echelon form or reduced row echelon form. This process helps to identify the pivot columns corresponding to the linearly independent vectors, which can be used to form a basis for the spanned space.

Transcript

00:01 In this example, we have five vectors provided, and what we'd like to do is define a vector space h to be the span of vectors v1, v2, v3, v4, and v5.

00:17 So now that h is the span of this set of vectors, one thing to notice is that the set here does not form a basis.

00:26 The reason being is that we have five vectors, but only four.

00:31 Entries per vector and whenever we have more vectors than entries per vector the set here would be linearly dependent so we know it spans and if we want a basis what we need to do is to decide which vectors to delete so that we still span h and form a basis and as you might have suspected the trick to determining all of those things at once is to form a mace tricks and row reduce so let's start off by saying let a matrix a be given as v1, v2, v3, v4, and v5 as its columns.

01:13 Then this matrix is going to be 1, 0, 0 ,0, negative 3, 0 ,0, 2, negative 3, negative 4, 1, 6, and 1, negative 3, negative 8, 7.

01:28 Last it's 2 -1 negative 6 and 9 so this is going to be possibly an interesting row reduction since we're dealing with a 4 by 5 matrix but let's see what we get into we're only going for echelom form let's remind ourselves and so from this pivot there's just 2 inches we need to eliminate this and this entry so here i've copied the first rows of this matrix and now let's multiply row 1 by positive 3 add the results to row 3.

02:02 What will obtain is a 0, 2, negative 8, negative 5, and 0.

02:12 For the next step, multiply row 1 by this time a negative 2, and then add the result to the last row.

02:20 This operation will give us altogether a 0, then negative 3, looks like a positive 12, then a 5, and a 5.

02:31 So let's analyze this new matrix.

02:33 We have a pivot here with zeros below.

02:36 We have a pivot here, it has a zero above but keep in mind we're just going for echelon, so that's nice but not needed.

02:43 But we need to eliminate the two and the negative three.

02:46 So let's start by recopping this matrix rows 1s and 2...