## Linear Algebra and Its Applications, Review Exercise 2.19

Review exercise 2.19. Consider the set of elementary 3 by 3 matrices $E_{ij}$ with ones on the diagonal and at most one nonzero entry below the diagonal. What subspace is spanned by these matrices?

Answer: An example member of this set is $E_{32} = \begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 0&1&1 \end{bmatrix}$

Since each matrix in the set has only zeros above the diagonal, a linear combination pf such matrices will also have only zeros above the diagonal.

Since each matrix in the set has ones on the diagonal, multiplying a given matrix by a scalar produces a matrix which has that scalar value for all diagonal entries. Adding a number of such matrices in turn produces a matrix for which all diagonal entries are equal.

Since each matrix in the set has some nonzero entry below the diagonal in some arbitrary location, a linear combination of such matrices could have any arbitrary set of values below the diagonal.

The subspace spanned by the set of 3 by 3 elementary matrices $E_{ij}$ is therefore the set of all 3 by 3 lower triangular matrices for which the diagonal values are equal to one another.

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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