## Linear Algebra and Its Applications, Exercise 3.1.16

Exercise 3.1.16. Describe the set of all vectors orthogonal to the vectors $(1, 4, 4, 1)$ and $(2, 9, 8, 2)$.

Answer: If a vector $x$ is orthogonal to the vectors $(1, 4, 4, 1)$ and $(2, 9, 8, 2)$ then its inner products with those vectors must be zero, so that $1 \cdot x_1 + 4 \cdot x_2 + 4 \cdot x_3 + 1 \cdot x_4 = 0$

and $2 \cdot x_1 + 9 \cdot x_2 + 8 \cdot x_3 + 2 \cdot x_4 = 0$

This is a system of two equations in four unknowns: $\begin{array}{rcrcrcrcl} x_1&+&4x_2&+&4x_3&+&x_4&=&0 \\ 2x_1&+&9x_2&+&8x_3&+&2x_4&=&0 \end{array}$

and is equivalent to the system $Ax = 0$ where $A = \begin{bmatrix} 1&4&4&1 \\ 2&9&8&2 \end{bmatrix}$

In other words, the problem of finding all vectors orthogonal to $(1, 4, 4, 1)$ and $(2, 9, 8, 2)$ is equivalent to the problem of finding the nullspace of the 2 by 4 matrix $A$ for which $(1, 4, 4, 1)$ and $(2, 9, 8, 2)$ are the rows.

We use Gaussian elimination to solve the system, starting by subtracting 2 times row 1 from row 2: $\begin{bmatrix} 1&4&4&1 \\ 2&9&8&2 \end{bmatrix} \Rightarrow \begin{bmatrix} 1&4&4&1 \\ 0&1&0&0 \end{bmatrix}$

The resulting echelon matrix has two pivots, in columns 1 and 2, so $x_1$ and $x_2$ are basic variables and $x_3$ and $x_4$ are free variables.

Setting $x_3 = 1$ and $x_4 = 0$, from row 2 we have $x_2 = 0$ and from row 1 we then have $x_1 + 4x_2 + 4x_3 + x_4 = x_1 + 0 + 4 + 0 = 0$ or $x_1 = -4$. So $(-4, 0, 1, 0)$ is one solution to $Ax = 0$. Setting $x_3 = 0$ and $x_4 = 1$, from row 2 we have $x_2 = 0$ again and from row 1 we have $x_1 + 4x_2 + 4x_3 + x_4 = x_1 + 0 + 0 + 1 = 0$ or $x_1 = -1$. So $(-1, 0, 0, 1)$ is a second solution to $Ax = 0$.

The vectors $(-4, 0, 1, 0)$ and $(-1, 0, 0, 1)$ together form a basis for the nullspace of $A$. That nullspace, i.e., all linear combinations of $(-4, 0, 1, 0)$ and $(-1, 0, 0, 1)$, is the set of all vectors orthogonal to the vectors $(1, 4, 4, 1)$ and $(2, 9, 8, 2)$.

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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