Q53E Find the basis of the space V聽o... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q53E (page 234)

Find the basis of the space Vof all skew-symmetric 3X3 matrices, and thus determine the dimension of V.

Short Answer

Expert verified

The dimension of a 3X3 skew-symmetric matrices is 6 which is spanned by .

$S p a n \{[\begin{matrix} 0 & 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ - 1 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{matrix}]\}$

Step by step solution

Determine the basis.

Consider the matrix $A = [\begin{matrix} 0 & a_{12} & a_{13} \\ - a_{12} & 0 & a_{23} \\ - a_{13} & - a_{3} & 0 \end{matrix}]$ where all $a_{i j}$ are real.

The matrix A is skew-symmetric if $A^{T} = - A$ and the general form of any skew-symmetric matrix is $[\begin{matrix} 0 & b & d \\ - b & 0 & e \\ - d & - e & 0 \end{matrix}]$ .

Simplify the equation $A = [\begin{matrix} 0 & a_{12} & a_{13} \\ - a_{12} & 0 & a_{23} \\ - a_{13} & - a_{3} & 0 \end{matrix}]$ as follows.

$A = [\begin{matrix} 0 & a_{12} & a_{13} \\ - a_{12} & 0 & a_{23} \\ - a_{13} & - a_{3} & 0 \end{matrix}] A = [\begin{matrix} 0 & a_{12} & 0 \\ - a_{12} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] + [\begin{matrix} 0 & 0 & a_{13} \\ 0 & 0 & 0 \\ - a_{13} & 0 & 0 \end{matrix}] + [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & a_{23} \\ 0 & - a_{23} & 0 \end{matrix}] A = a_{12} [\begin{matrix} 0 & 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] + a_{13} [\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ - 1 & 0 & 0 \end{matrix}] + a_{23} [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{matrix}] where [\begin{matrix} 0 & 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ - 1 & 0 & 0 \end{matrix}] and [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{matrix}] are linear independent .$

Hence, the dimension of A is 3 and spanned by .

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91影视

Find the basis of the space Vof all skew-symmetric 3X3 matrices, and thus determine the dimension of V.

Short Answer

Step by step solution

Determine the basis.

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