## Halmos Sections 32 and 33: Linear Transformations and Transformations as Vectors (Pt. I)

**Point of post: **In this post we complete the problems that appear at the end of Halmos, sections 32 and 33.

1.

**Problem: ** Prove that each of the correspondences described below is a linear transformation.

**a) ** where is considered a real vector space.

**b) **

**c) **

**d) **

**e) **

**f) **

*Remark: *If the notation seems unfamiliar, that’s because I created my notation. See here and here

**Proof:**

**a) **We note that

*Remark: *It was crucial in the above that since otherwise we might not be able to claim that the imaginary part was the sum of the original imaginary parts. For example, it’s not true that .

**b) **For this we merely note that

and

from where the conclusion follows.

**c) **To do this merely note that for we have that

and

And thus since every element of can be written as a linear combination of vectors of the form it follows that is additive, from where the conclusion follows.

**d) **This is fairly similar, namely

and

**e) **For this we merely note that by part **d) **

**f) **Very similarly, using part **d) **again

2. **Problem: **Prove that is a vector space. What’s its dimension?

**Proof: **For the notation and proof see here

**References:**

1. Halmos, Paul R. *Finite-dimensional Vector Spaces,*. New York: Springer-Verlag, 1974. Print

[…] of post: This is a continuation of this post in an effort to answer the questions at the end of sections 32 and 33 in Halmos’s […]

Pingback by Halmos Sections 32 and 33: Linear Transformations and Transformations as Vectors (Pt. II) « Abstract Nonsense | November 22, 2010 |

Can you clearly post solution for exercise 2 of this section? Thank you!

Comment by Amira | November 29, 2010 |

If you’ll kindly follow the link, I prove a more general statement there. Is that not satisfactory?

Comment by drexel28 | November 29, 2010 |