## Linear Algebra and Its Applications, Review Exercise 2.10

Review exercise 2.10. Given the set of all linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^n$ define operations for scalar multiplication and vector addition that will make the set a vector space. What is the dimension of the resulting vector space?

Answer: Let $A$ and $B$ be linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^n$ and let $[A]$ and $[B]$ be the $n$ by $n$ matrices corresponding to $A$ and $B$ respectively. If $c$ is a scalar define the scalar product $cA$ to be the linear transformation represented by the matrix $c[A]$ and define the vector sum $A+B$ to be the linear transformation represented by the matrix $[A]+[B]$.

The set of linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^n$ is a vector space under the operations thus defined; this follows from the fact that the set of $n$ x $n$ matrices is a vector space under those operations. The dimension of the space is $n^2$.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

This entry was posted in linear algebra. Bookmark the permalink.

### 2 Responses to Linear Algebra and Its Applications, Review Exercise 2.10

1. Math Nerd says:

\mathbb{R}^n to \mathbb{R}^n , should the dimensions not be n?

• hecker says:

First, thanks for reading the blog and commenting! Now, to address your question: The key point is that the linear transformations are represented as n by n matrices: Any linear transformation from R^n to R^n can be represented as an n by n matrix, and any n by n matrix represents a linear transformation from R^n to R^n. The two sets are therefore equivalent (or isomorphic, to use the technical term). The set of n by n matrices is a vector space (because it obeys the relevant rules for scalar multiplication and vector addition) and thus the space of linear transformations is also a vector space.

Now consider an n by n matrix. Every such matrix can be alternatively presented as a vector in the space R^(n^2): Take the first row of the matrix and put its elements in positions 1 through n of the vector, take the second row of the matrix and put its elements in positions n+1 through n+n of the vector, and so on. The final vector (containing all elements of the matrix) will have length n x n or n^2.

Since any n by n matrix can be represented as a vector of length n^2 (and vice versa), the dimension of the space of n by n matrices is the same as the dimension of R^(n^2), namely n^2. And since the space of n by n matrices maps one-to-one onto the space of linear transformations from R^n to R^n, the dimension of the space of linear transformations is also n^2.