A basis 𝐯1, …, 𝐯n of ℝ^n is called right-handed if the n ×n matrix A=(𝐯1 𝐯2 …𝐯n) whose columns a

step 1 Chapter 2, exercise 48 of section 6. Suppose vb is equal to B sub 1, v .1, plus B sub 2, v .2, e

A basis 𝐯1, …, 𝐯n of ℝ^n is called right-handed if the n ×n...

A basis $\mathbf{v}_1, \ldots, \mathbf{v}_n$ of $\mathbb{R}^n$ is called right-handed if the $n \times n$ matrix $A=\left(\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_n\right)$ whose columns are the basis vectors has positive determinant: $\operatorname{det} A>0$. If $\operatorname{det} A<0$, the basis is called left-handed. (a) Which of the following form right-handed bases of $\mathbb{R}^3$ ? (i) $\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 0\end{array}\right)$, (ii) $\left(\begin{array}{l}2 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 1 \\ 2\end{array}\right)$ (iii) $\left(\begin{array}{r}-1 \\ 2 \\ 3\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ -2\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ 2\end{array}\right)$ (iv) $\left(\begin{array}{l}3 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right),\left(\begin{array}{l}2 \\ 1 \\ 3\end{array}\right)$. (b) Show that if $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ is a left-handed basis of $\mathbb{R}^3$, then $\mathbf{v}_2$, $\mathbf{v}_1, \mathbf{v}_3$ and $-\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are both right-handed bases. (c) What sort of basis has $\operatorname{det} A=0$ ?

A basis $\mathbf{v}_1, \ldots, \mathbf{v}_n$ of $\mathbb{R}^n$ is called right-handed if the $n \times n$ matrix $A=\left(\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_n\right)$ whose columns are the basis vectors has positive determinant: $\operatorname{det} A>0$. If $\operatorname{det} A<0$, the basis is called left-handed. (a) Which of the following form right-handed bases of $\mathbb{R}^3$ ?
(i) $\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 0\end{array}\right)$,
(ii) $\left(\begin{array}{l}2 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 1 \\ 2\end{array}\right)$
(iii) $\left(\begin{array}{r}-1 \\ 2 \\ 3\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ -2\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ 2\end{array}\right)$ (iv) $\left(\begin{array}{l}3 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right),\left(\begin{array}{l}2 \\ 1 \\ 3\end{array}\right)$.
(b) Show that if $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ is a left-handed basis of $\mathbb{R}^3$, then $\mathbf{v}_2$, $\mathbf{v}_1, \mathbf{v}_3$ and $-\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are both right-handed bases. (c) What sort of basis has $\operatorname{det} A=0$ ?

Key Concepts

Basis in R^n

A basis for ?? is a collection of n linearly independent vectors that span the entire space. Every vector in the space can be uniquely represented as a linear combination of these basis vectors, which establishes a coordinate framework for analyzing and manipulating the space.

Determinant and Orientation

The determinant of a matrix formed by placing basis vectors as columns gives a measure of the volume of the parallelepiped spanned by these vectors, with the sign of the determinant indicating the orientation. A positive determinant implies one orientation, often associated with the standard or 'right-handed' orientation, while a negative determinant indicates the opposite, or 'left-handed', orientation.

Right-Handed and Left-Handed Bases

A right-handed basis is one in which the determinant of the matrix with these vectors as columns is positive, meaning that the basis preserves the standard orientation of the space. A left-handed basis has a negative determinant, signifying a reversal of this orientation.

Effect of Basis Reordering on Determinant

Changing the order of basis vectors, such as swapping two vectors, changes the sign of the determinant. This fact is crucial because it means that rearranging a left-handed basis (with a negative determinant) by interchanging the positions of specific vectors can yield a right-handed basis (with a positive determinant), while preserving the span of the space.

Degenerate Basis and Zero Determinant

If the determinant of the matrix formed by a set of vectors is zero, it implies that the vectors are linearly dependent and hence do not form a basis for ??. A zero determinant indicates that the vectors lie within a lower-dimensional subspace, resulting in zero volume for the supposed parallelepiped.

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