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.

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

Math Nerd says:

November 9, 2014 at 5:58 am

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

- hecker says:
  
  November 9, 2014 at 4:36 pm
  
  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.