Question

Linear Algebra:Let A be an nxn matrix with real entries Is the set {X vector space ? A detailed justification of your answer is required. nxn matrix with real entries AX=xAſ a а

Answer 1

Explanation:

Here A be n×n matrix with real enties

y= Ax=xA is vector space.

Let m be set of all n*n matrix with real enties then m is vector space over IR.

we show y is vector subspace of m.

Here
`I_(n*n\\)` identity matrix

IA=AI

∴ I ∈ y

∴ y is non empty subset of m.

Also if
`x_(1)`,
`x_(2)` ∈ y ⇒ A
`x_(1)`=
`x_(1)`A ,A
`x_(2)`=
`x_(2)`A

for
`\alpha` ∈ IR arbitrary

`(\alpha x_(1) +x_(2) )A=\alpha (x_(1)A)+x_(2) A\\=\alpha (Ax_(1))+Ax_(2)\\ =A(\alpha x_(1) +x_(2))\\`

Hence
`\alpha x_(1)+x_(2)` ∈ y ∀
`x_(1),x_(2)` ∈ y

∴ y is subspace of m.

∴ y is vector space.