## Linear Algebra and Its Applications, Exercise 2.4.10

Exercise 2.4.10. Suppose the nullspace of a matrix is the set of all vectors $x = \begin{bmatrix} x_1& x_2&x_3 \end{bmatrix}^T$ in $\mathbf R^3$ for which $x_1 + 2x_2 + 4x_3 = 0$. Find a 1 by 3 matrix $A$ with this nullspace. Find a 3 by 3 matrix $A'$ with the same nullspace.

Answer: If $A = \begin{bmatrix} a_1&a_2&a_3 \end{bmatrix}$ and $Ax = 0$ then we have $a_1x_1 + a_2x_2 + a_3x_3 = 0$. We know that $x_1 + 2x_2 + 4x_3 = 0$ so one way to construct a suitable matrix $A$ is to set $a_1 = 1$, $a_2 = 2$, and $a_3 = 4$ so that $A = \begin{bmatrix} 1&2&4 \end{bmatrix}$.

For a 3 by 3 matrix $A'$, if $A'x = 0$  each row of $A'$ times $x$ is 0 so that $a_{i1}x_1 + a_{i2}x_2 + a_{i3}x_3 = 0$ for $1 \le i \le 3$. Since $x_1 + 2x_2 + 4x_3 = 0$ we can construct a suitable matrix $A'$ by setting each row of $A'$ to $\begin{bmatrix} 1&2&4 \end{bmatrix}$ or to any multiple of that vector. For example, one possible choice of $A'$ is $A' = \begin{bmatrix} 1&2&4 \\ 2&4&8 \\ 4&8&16 \end{bmatrix}$

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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### 2 Responses to Linear Algebra and Its Applications, Exercise 2.4.10

1. Kenny says:

Hi hecker, while we doing the A’ 3X3 matrix, since the information that x1+2×2+4×3=0 is given, can we be sure that A’ matrix is a rank1 matrix??

• hecker says:

A’ has rank 1 because the second and third columns are multiples of the first column, and thus are linearly dependent on the first column. The rank of a matrix is the number of linearly independent columns, and in this case there is only 1 such column.