Finding the Basis of a Vector Space

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  • 0:02 Vector Space and Basis
  • 1:02 Reduced Row Echelon Form
  • 2:05 Finding the Basis
  • 3:03 Lesson Summary
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Lesson Transcript
Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

In this lesson we'll start by reviewing matrix reduced row echelon form, which is integral to finding a basis of a vector space. Then we'll work through a problem together to see exactly how finding a basis is accomplished.

Vector Space and Basis

In math, we often work with sets, or collections, of expressions. For example, you could have an ordered set of numbers (a sequence) and have a problem that tells you to find the next number in the sequence. Another common example is working with a set of equations to solve the variables in them.

In linear algebra, you might find yourself working with a set of vectors. When the operations of scalar multiplication and vector addition hold for a set of vectors, we call it a vector space. When working with a vector space, one thing you might want to do is identify the vectors that form a basis for it. A vector space's basis is a subset of vectors within the space that are linearly independent and span the space.

A basis is linearly independent because the vectors in it cannot be defined as a linear combination of any of the other vectors in the basis. By spanning the vector space, we mean that the vectors in that space can be defined as a linear combination of the vectors in the basis.

Reduced Row Echelon Form

Before we go on with vectors, we need to do a quick review of matrix operations. The most essential step to finding the basis of a vector space actually involves a matrix. More specifically, you'll need to be able to put a matrix in reduced row echelon form, which adheres to the following four conditions:

  1. All non-zero rows are above any rows containing only zeros.
  2. The first non-zero entry in a row is to the right of the first non-zero entry of the row above it.
  3. All elements in the same column as the first non-zero element in a row are zeroes.
  4. The first non-zero entry of each row is a one.

In order to get a matrix in reduced row echelon form, there are three different operations that can be performed on the rows of a matrix.

  1. You can swap any two rows within a matrix.
  2. A row in the matrix can be multiplied by a constant.
  3. A row can have a multiple of another row added to it.

The process of using these three operations to get a matrix into reduced row echelon form is called row reduction.

Finding the Basis

Now we have everything we need to find a basis of a vector space. Let's work through an example problem. Note that vector spaces can have multiple bases, and here we will find one possible basis for the following vector space.

vector space

In order to find a basis for this vector space, we start by putting these vectors into the columns of a matrix.

vector space matrix

Once this is done, we use the row reduction operations we just went over to put the matrix into reduced row echelon form. This will be the most mathematically intensive step in finding the basis of a vector space.

row reduction steps for finding reduced row echelon form

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