Find a subset of the given vectors that forms a basis for...

Find a subset of the given vectors that forms a basis for the space spanned by those vectors, and then express each vector that is not in the basis as a linear combination of the basis vectors. $$\begin{array}{l} \mathrm{v}_{1}=(1,-1,5,2), \mathrm{v}_{2}=(-2,3,1,0) \\ \mathrm{v}_{3}=(4,-5,9,4), \mathrm{v}_{4}=(0,4,2,-3) \\ \mathrm{v}_{5}=(-7,18,2,-8) \end{array}$$

Find a subset of the given vectors that forms a basis for the space spanned by those vectors, and then express each vector that is not in the basis as a linear combination of the basis vectors.
$$\begin{array}{l}
\mathrm{v}_{1}=(1,-1,5,2), \mathrm{v}_{2}=(-2,3,1,0) \\
\mathrm{v}_{3}=(4,-5,9,4), \mathrm{v}_{4}=(0,4,2,-3) \\
\mathrm{v}_{5}=(-7,18,2,-8)
\end{array}$$

Key Concepts

Gaussian Elimination

Gaussian elimination, or row reduction, is a systematic algorithm for simplifying matrices. It is used to determine the pivot positions which help identify linearly independent vectors in a set, to find a basis for the span, and to solve systems of linear equations.

Linear Independence

Linear independence is a property of a set of vectors where no vector in the set can be expressed as a linear combination of the others. This concept is critical for ensuring that a basis is efficient and unique for representing elements of the vector space.

Linear Combination

A linear combination involves multiplying vectors by scalars and then summing the results. It is the fundamental operation used to construct new vectors from existing ones and is central to understanding how vectors relate in terms of span and basis.

Spanning Set

A spanning set of a vector space is a collection of vectors from which any vector in the space can be expressed as a linear combination. The span of a set defines the subspace consisting of all such linear combinations.

Vector Spaces

A vector space is a set of vectors along with operations of vector addition and scalar multiplication that satisfy specific axioms. It provides the framework in which concepts like linear combinations, span, and basis are defined and analyzed.

Basis

A basis of a vector space (or of a subspace) is a minimal set of linearly independent vectors that span the space. It provides the simplest coordinate system for representing every vector in the space uniquely as a linear combination of the basis vectors.

Transcript

00:01 So the way we find the basis of the span of the set of vectors that given here, v1 through v5, is to put the vectors into a matrix and then find the column space of the matrix.

00:11 So we have the vectors here to find the column space we start doing gauzing elimination.

00:16 So we copy over our first row, and then we're going to add the first two rows together to get our new second row.

00:28 So we get a 0, 1, a negative 1, a 4, and an 11.

00:34 Then we're going to multiply our first row by negative.

00:37 5 and added to our third row.

00:39 So that's going to be a 0.

00:41 Negative 2 times negative 5 is 10.

00:42 So this is going to be an 11.

00:44 Negative 4 times negative 5 is negative 20 plus 9 is negative 11.

00:51 Then we just have a 2 here.

00:53 Then we have negative 7 times negative 5, which is going to be 35 plus 2 is just 37.

01:03 And then we'll multiply our top row by negative 2 and add it to the last row.

01:07 So we get a 0.

01:08 We get a positive 4 here.

01:11 We get a negative 4 here, a negative 3 here.

01:15 And then we get negative 7 times negative 2, which is positive 14, minus 8, which is going to be just a 6.

01:23 All right, so let's go ahead and copy that over to the left.

01:28 And then we'll try to zero out the second column.

01:30 We'll do that by multiplying the second row by 2 and adding it to the first.

01:34 So that gives us a 1, a 0, a 2, a 2, and 8...

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