## Linear Algebra and Its Applications, Review Exercise 2.23

Review exercise 2.23. Given any three vectors $v_1$, $v_2$, and $v_3$ in $\mathbb{R}^3$ find a matrix $A$ that transforms the three elementary vectors $e_1$, $e_2$, and $e_3$ respectively into those three vectors.

Answer: When multiplying $e_1$ by $A$ only the entries in the first column of $A$ are multiplied by one; all other entries are multiplied by zero. So the first column of $A$ should be set to $v_1$. Similarly when multiplying $e_2$ by $A$ only the entries in the second column of $A$ are multiplied by one; all other entries are multiplied by zero. The second column of  $A$ should therefore be set to $v_2$. Finally the third column of $A$ should be set to $v_3$.

If $v_1 = (a_1, a_2, a_3)$, $v_2 = (b_1, b_2, b_3)$, and $v_3 = (c_1, c_2, c_3)$ then we have

$A = \begin{bmatrix} a_1&b_1&c_1 \\ a_2&b_2&c_2 \\ a_3&b_3&c_3 \end{bmatrix}$

so that (for example)

$Ae_2 = \begin{bmatrix} a_1&b_1&c_1 \\ a_2&b_2&c_2 \\ a_3&b_3&c_3 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$

$= \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = v_2$

Note that if $A$ is invertible if and only if the three vectors $v_1$, $v_2$, and $v_3$ are linearly independent.

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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