## Linear Algebra and Its Applications, Exercise 2.1.8

Exercise 2.1.8. Consider the following system of linear equations: $Ax = \begin{bmatrix} 1&1&1 \\ 1&0&2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$

Does the set of solutions $x$ form a point, line, or plane? Is it a subspace? Is it the nullspace of $A$? The column space of $A$?

Answer: The system $Ax = 0$ corresponds to $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}x_1&+&x_2&+&x_3&=&0 \\ x_1&&&+&2x_3&=&0 \end{array}$

From the second equation we have $x_1 = -2x_3$ and we can substitute into the first equation to obtain $-2x_3 + x_2 + x_3 = 0$ or $x_2 - x_3 = 0$. We therefore have $x_2 = x_3$ so that the set of solutions $x$ can be represented as $x = \begin{bmatrix} -2x_3 \\ x_3 \\ x_3 \end{bmatrix} = x_3 \begin{bmatrix} -2 \\ 1 \\ 1 \end{bmatrix}$

where $x_3$ is a free variable. The set of solutions $x$ is therefore a line passing through the origin and the point $(-2, 1, 1)$.

Since the solution set is a line passing through the origin it is a subspace, and since $x$ is the set of all vectors for which $Ax = 0$ it is by definition the nullspace $\mathcal{N}(A)$ of $A$ (see page 68).

The column space of $A$ is the set of all vectors $v$ that are linear combinations of the columns of $A$: $v = c_1 \begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_2 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_3 \begin{bmatrix} 1 \\ 2 \end{bmatrix}$

and thus contains 2 by 1 vectors. All solution vectors $x$ are 3 by 1 vectors; the set of solutions $x$ is not the same as the column space of $A$.

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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