Question

skew-symmetric 3 x 3 matrices form as subspace of all 3 x 3 matrices and find a basis for this subspace.

Answer 1

a) ∝A ∈ W

so by subspace, W is subspace of 3 × 3 matrix

b) therefore Basis of W is

={
`{\left[\begin{array}{ccc}0&1&0\\-1&0&0\\0&0&0\end{array}\right]`
`,\left[\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}\right]`
`,\left[\begin{array}{ccc}0&0&0\\0&0&1\\0&-1&0\end{array}\right]`}

Explanation:

Given the data in the question;

W = 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 =
`\left[\begin{array}{ccc}o&a&b\\-a&o&c\\-b&-c&0\end{array}\right]`

=
`a\left[\begin{array}{ccc}0&1&0\\-1&0&0\\0&0&0\end{array}\right]`
`+b\left[\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}\right]`
`+c\left[\begin{array}{ccc}0&0&0\\0&0&1\\0&-1&0\end{array}\right]`

therefore Basis of W is

={
`{\left[\begin{array}{ccc}0&1&0\\-1&0&0\\0&0&0\end{array}\right]`
`,\left[\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}\right]`
`,\left[\begin{array}{ccc}0&0&0\\0&0&1\\0&-1&0\end{array}\right]`}