Answer:
a) ∝A ∈ W
so by subspace, W is subspace of 3 × 3 matrix
b) therefore Basis of W is
={ [tex]{\left[\begin{array}{ccc}0&1&0\\-1&0&0\\0&0&0\end{array}\right][/tex] [tex],\left[\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}\right][/tex] [tex],\left[\begin{array}{ccc}0&0&0\\0&0&1\\0&-1&0\end{array}\right][/tex]}
Step-by-step explanation:
Given the data in the question;
W = { A| Air Skew symmetric matrix}
= {A | A = -A^T }
A ; O⁻ = -O⁻^T O⁻ : Zero mstrix
O⁻ ∈ W
now let A, B ∈ W
A = -A^T B = -B^T
(A+B)^T = A^T + B^T
= -A - B
- ( A + B )
⇒ A + B = -( A + B)^T
∴ A + B ∈ W.
∝ ∈ | R
(∝.A)^T = ∝A^T
= ∝( -A)
= -( ∝A)
(∝A) = -( ∝A)^T
∴ ∝A ∈ W
so by subspace, W is subspace of 3 × 3 matrix
A ∈ W
A = -AT
A = [tex]\left[\begin{array}{ccc}o&a&b\\-a&o&c\\-b&-c&0\end{array}\right][/tex]
= [tex]a\left[\begin{array}{ccc}0&1&0\\-1&0&0\\0&0&0\end{array}\right][/tex] [tex]+b\left[\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}\right][/tex] [tex]+c\left[\begin{array}{ccc}0&0&0\\0&0&1\\0&-1&0\end{array}\right][/tex]
therefore Basis of W is
={ [tex]{\left[\begin{array}{ccc}0&1&0\\-1&0&0\\0&0&0\end{array}\right][/tex] [tex],\left[\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}\right][/tex] [tex],\left[\begin{array}{ccc}0&0&0\\0&0&1\\0&-1&0\end{array}\right][/tex]}
