verify The set of all 2 × 2 invertible matrices with the standard matrix addition and scalar multiplication is a vector space or not?

Answer 1

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The set of all invertible n×nn×n matrices is not a vector space with respect to the typical matrix addition and scalar multiplication operations and the typical matrix zero.

However,

The special orthogonal group (rotation matrices) is a vector space if you use matrix multiplication for the addition operator and the identity matrix as the zero matrix.

You also have to make scalar multiplication exponentials.

You can also change the zero matrix to be a matrix CC with:

A⊕B=ABC−1A⊕B=ABC−1
and

s⊗A=AsC−ss⊗A=AsC−s.

Then,

A⊕C=AA⊕C=A.

And invertible n×nn×n matrices are just rotation matrices with a scaling part and so are also vector spaces.

its very trickyy but easy