Determine which of the following are bases of ℝ^3 : (a) ( 2 1 5 ),( 1

step 1 All right, we're looking at chapter 2, section 3, question 44A. All right, so this is not a basi

Determine which of the following are bases of ℝ^3 : (a) ( ...

Determine which of the following are bases of $\mathbb{R}^3$ : (a) $\left(\begin{array}{l}2 \\ 1 \\ 5\end{array}\right),\left(\begin{array}{l}1 \\ 5 \\ 2\end{array}\right)$ (b) $\left(\begin{array}{r}0 \\ 1 \\ -5\end{array}\right)$, $$ \left(\begin{array}{r} -1 \\ 3 \\ 0 \end{array}\right),\left(\begin{array}{l} 1 \\ 3 \\ 0 \end{array}\right) $$ (c) $$ \left(\begin{array}{r} 0 \\ 4 \\ -1 \end{array}\right),\left(\begin{array}{r} -1 \\ 0 \\ 1 \end{array}\right),\left(\begin{array}{r} 1 \\ -8 \\ 1 \end{array}\right) $$ $$ \left(\begin{array}{r} 2 \\ 0 \\ -2 \end{array}\right),\left(\begin{array}{r} -1 \\ 2 \\ -1 \end{array}\right),\left(\begin{array}{r} 0 \\ -1 \\ 0 \end{array}\right),\left(\begin{array}{r} -1 \\ 2 \\ 1 \end{array}\right) \text {. } $$

Determine which of the following are bases of $\mathbb{R}^3$ :
(a) $\left(\begin{array}{l}2 \\ 1 \\ 5\end{array}\right),\left(\begin{array}{l}1 \\ 5 \\ 2\end{array}\right)$
(b) $\left(\begin{array}{r}0 \\ 1 \\ -5\end{array}\right)$,
$$
\left(\begin{array}{r}
-1 \\
3 \\
0
\end{array}\right),\left(\begin{array}{l}
1 \\
3 \\
0
\end{array}\right)
$$
(c)
$$
\left(\begin{array}{r}
0 \\
4 \\
-1
\end{array}\right),\left(\begin{array}{r}
-1 \\
0 \\
1
\end{array}\right),\left(\begin{array}{r}
1 \\
-8 \\
1
\end{array}\right)
$$
$$
\left(\begin{array}{r}
2 \\
0 \\
-2
\end{array}\right),\left(\begin{array}{r}
-1 \\
2 \\
-1
\end{array}\right),\left(\begin{array}{r}
0 \\
-1 \\
0
\end{array}\right),\left(\begin{array}{r}
-1 \\
2 \\
1
\end{array}\right) \text {. }
$$

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Key Concepts

Basis

A basis of a vector space is a set of vectors that is both linearly independent and spans the entire space. In the context of ?³, a basis must consist of three vectors such that any vector in ?³ can be written as a linear combination of these vectors.

Linear Independence

Linear independence refers to the property that no vector in a set can be expressed as a linear combination of the others. For vectors in ?³, having three vectors that are linearly independent ensures they do not lie in the same plane or line, which is critical for forming a basis.

Span

The span of a set of vectors is the collection of all possible linear combinations of those vectors. For a set in ?³ to be a basis, the span of the vectors must equal ?³, meaning every vector in ?³ can be generated from these basis vectors.

Dimension

The dimension of a vector space, such as ?³, is the number of vectors in any basis for the space. Since ?³ has a dimension of 3, any basis must consist of exactly three linearly independent vectors.

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