(a) Show that ( 4 0 1 ),( 2 1 0 ), and ( 2 -1 1

step 1 For exercise 32, part A, we have our basis of 1 -0 -0 -1 -0 -1 -0 -1 and 0 -0 -1. Then for B, th

(a) Show that ( 4 0 1 ),( 2 1 ...

(a) Show that $\left(\begin{array}{l}4 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}2 \\ 1 \\ 0\end{array}\right)$, and $\left(\begin{array}{r}2 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{r}0 \\ 2 \\ -1\end{array}\right)$ are two different bases for the plane $x-2 y-4 z=0$. (b) Show how to write both elements of the second basis as linear combinations of the first. (c) Can you find a third basis?

Key Concepts

Vector Space and Subspace

A vector space is a collection of objects called vectors that can be added together and scaled by numbers, known as scalars. A subspace is a subset of a vector space that is itself a vector space under the same operations. In this context, the plane defined by the equation is a 2-dimensional subspace of three?dimensional space, and understanding subspaces is crucial when discussing bases and spans.

Basis

A basis for a vector space (or subspace) is a set of vectors that is both linearly independent and spans the space. This means any vector in the space can be uniquely expressed as a linear combination of the basis vectors. Identifying different bases for the same subspace involves verifying these properties with different sets of vectors.

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. Establishing that a set of vectors is linearly independent is essential when determining if they form a basis, as it guarantees that the representation of any vector in the space is unique.

Span

The span of a set of vectors is the collection of all possible linear combinations of those vectors. When we say a set of vectors spans a space (or subspace), we mean that every vector in that space can be generated by some linear combination of the set. This concept is fundamental to verifying that a collection of vectors makes up a basis.

Linear Combination and Change of Basis

Expressing vectors as linear combinations of other vectors is a key technique in linear algebra, particularly when shifting from one basis to another—a process known as change of basis. This involves finding the coefficients that link vectors in one basis with the corresponding representation in another basis. Understanding this process is vital for performing coordinate transformations and analyzing different representations of the same geometric objects.

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