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Chapter 4: Problem 14

Explain why \(S\) is not a basis for \(R^{2}\) $$S=\\{(-1,2)\\}$$

Short Answer

Expert verified
The set \(S = \{(-1,2)\}\) cannot be a basis for \(R^{2}\) because it only contains one vector, and a basis for \(R^{2}\) requires at least two vectors.

Step by step solution

01

Definition of a Basis

In linear algebra, a basis for a vector space is a set of vectors that span the space and are linearly independent. This means that there should be at least two vectors in \(S\) to form a basis for \(R^{2}\).
02

Number of Vectors in Set S

In this case, set \(S\) only contains one vector, namely \((-1,2)\).
03

Determine if S is a Basis for \(R^{2}\)

Because a basis for \(R^{2}\) requires at least two vectors and set \(S\) only contains one vector, \(S\) cannot form a basis for \(R^{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Basis of a Vector Space
In linear algebra, the concept of a "basis" is fundamental for understanding the structure of a vector space. A basis of a vector space is a set of vectors that fulfill two important criteria: they must be linearly independent, and they must span the vector space. This means that any vector in the space can be expressed as a unique combination of the basis vectors. For example, a basis for the vector space \(\mathbb{R}^2\) requires exactly two vectors. These vectors define a plane where any other vector in \(\mathbb{R}^2\) can be created by scaling and adding the basis vectors. Without meeting these conditions, a set cannot be termed as a basis.
Linear Independence
Linear independence is a crucial concept when talking about the "basis of a vector space." A set of vectors is said to be linearly independent if no vector in the set can be written as a linear combination of others. In other words, each vector adds something new and is not redundant.

In the context of \(\mathbb{R}^2\), any two vectors are independent if neither is a scalar multiple of the other. This independence ensures that the vectors can span the two-dimensional space. For our specific example of set \(S = \{(-1,2)\}\), there's only one vector, and thus, it fails to exhibit independence among two vectors as required for \(\mathbb{R}^2\).
Spanning Set
A spanning set in a vector space is a collection of vectors from which we can construct any vector in that space. To "span" a vector space means that a combination of the vectors in the set can describe every vector in the space.

In the case of \(\mathbb{R}^2\), two non-collinear vectors are needed to span the entire space. For the given set \(S = \{(-1,2)\}\), it can't span \(\mathbb{R}^2\) because we need at least two vectors to cover all possibilities in this two-dimensional space. The single vector can only describe a line, but not the entire plane. Therefore, it fails the spanning set criterion.

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