## 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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