Linear Algebra, 2020a

Section I. Complex Vector Spaces 401
With these rules, all of the operations that we’ve used for real vector spaces
carry over unchanged to vector spaces with complex scalars.
2.2 Example Matrix multiplication is the same, although the scalar arithmetic
involves more bookkeeping.
(
)(
)
1 + 1i 2 − 0i 1 + 0i 1 − 0i
i
(
−2 + 3i 3i −i
)
(1 + 1i) · (1 + 0i)+(2 − 0i) · (3i) (1 + 1i) · (1 − 0i)+(2 − 0i) · (−i)
=
(i) · (1 + 0i)+(−2 + 3i) · (3i) (i) · (1 − 0i)+(−2 + 3i) · (−i)
(
)
=
1 + 7i 1 − 1i
−9 − 5i 3 + 3i
We shall carry over unchanged from the previous chapters everything that
we can. For instance, we shall call this
⎛ ⎞ ⎛ ⎞
1 + 0i 0 + 0i
0 + 0i
〈
⎜
⎝
⎟
. ⎠ ,..., 0 + 0i
⎜
⎝
⎟
. ⎠ 〉
0 + 0i 1 + 0i
the standard basis for C n as a vector space over C and again denote it E n .
Another example is that P n will be the vector space of degree n polynomials
with coefficients that are complex.