Find a basis for the set of vectors in ℝ^3 in the plane x+2 y+z=0 .[ Hint: Think of the equation

Find a basis for the set of vectors in ℝ^3 in the plane x+2...

Key Concepts

Homogeneous Systems of Linear Equations

A homogeneous system is a system of linear equations in which all the constant terms are zero. Such systems always have at least the trivial solution (the zero vector), and the set of all solutions forms a vector subspace. Understanding homogeneous systems is essential to parameterize the solution set and identify its basis.

Vector Subspace

A vector subspace is a subset of a vector space that is itself a vector space under the operations of vector addition and scalar multiplication. In the context of systems of equations, the set of all solutions to a homogeneous equation forms a subspace, which means that any linear combination of solutions is also a solution.

Basis of a Vector Space

A basis of a vector space is a set of linearly independent vectors that span the entire space. In other words, every vector in the space can be expressed uniquely as a linear combination of the basis vectors. When finding a basis for the solution set of a homogeneous system, one typically finds a minimal set of independent vectors that generate all possible solutions.

Parameterization of Solution Sets

Parameterization involves expressing the solution of a system of equations in terms of free variables. This method allows one to describe the infinite set of solutions as linear combinations of vectors multiplied by free parameters, which directly leads to identifying a basis for the solution space.

Linear Independence

Linear independence is a property of a set of vectors where no vector in the set can be written as a linear combination of the others. This concept is crucial when selecting a basis, ensuring that the chosen vectors are the minimal necessary components to span the subspace without redundancy.

Transcript

00:01 In this example, we're dealing with the equation of a plane, which is x plus 2y plus z equals 0.

00:07 The most important thing about this plane is that the right -hand side is zero here.

00:12 That means, in particular, that this plane is going to go through the origin.

00:17 And we know now that planes in r3 that go through the origin are considered subspaces in r3 or of r3.

00:25 So in this case, what we might ask is, how could we get a basis for this subspace? we'll start off by taking the equation and solving for the variable x.

00:34 We'll have x is equal to negative 2y minus z.

00:39 Then what are the restrictions on the variables y and z? well, none are given so we could write y equals y if we like n z equals z to convey that there's no restrictions.

00:53 And this also helps remind us that y and z are also considered free variables in this situation.

00:58 Then with this setup, we can describe a generic vector x that is inside this plane, x plus 2y plus z equals 0.

01:08 X is going to be consisting of x, y, and z.

01:12 So let me put an arrow here to consider this as a vector and to differentiate it from this x.

01:18 Then the vector x is going to be described in terms of its free variables, which are y and z.

01:24 So let's write first that we have a vector y, multiplying a constant vector, plus a vector z, multiplying a different constant vector.

01:36 And the way we determine the constants is we take coefficients for our y to place here, which are negative 2, 1, and 0, since y is missing here.

01:46 Then take coefficients of z.

01:48 They are negative 1...

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