## 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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