Linear Algebra and Its Applications, Exercise 2.3.10

Exercise 2.3.10. The set of all 2 by 2 matrices forms a vector space under the standard rules for multiplying two matrices and multiplying a matrix by a scalar. Find a basis for the space and describe the subspace spanned by the set of all matrices U in echelon form.

Answer: Any 2 by 2 matrix

A = \begin{bmatrix} a&b \\ c&d \end{bmatrix}

can be represented as a linear combination of four matrices as follows:

A = \begin{bmatrix} a&b \\ c&d \end{bmatrix} = a \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix} + b \begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix} + c \begin{bmatrix} 0&0 \\ 1&0 \end{bmatrix} + d \begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix}

These four matrices are linearly independent and form a basis for the space of all 2 by 2 matrices.

The set of 2 by 2 echelon matrices consists of all matrices U of the form

\begin{bmatrix} a&b \\ 0&d \end{bmatrix}

where any of a, b, or d may be zero. The subspace spanned is the set of all upper triangular matrices, and the three matrices

\begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix} \qquad \begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix} \qquad \begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix}

serve as a basis for the subspace.

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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