Linear Algebra and Its Applications, Exercise 3.1.12

Exercise 3.1.12. For the matrix

A = \begin{bmatrix} 1&0&2 \\ 1&1&4 \end{bmatrix}

find a basis for the nullspace and show that it is orthogonal to the row space. Take the vector x = (3, 3, 3) and express it as the sum of a nullspace component x_n and a row space component x_r.

Answer: We use Gaussian elimination to solve the system Ax = 0. We start by subtracting 1 times row 1 from row 2:

\begin{bmatrix} 1&0&2 \\ 1&1&4 \end{bmatrix} \Rightarrow \begin{bmatrix} 1&0&2 \\ 0&1&2 \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 is a free variable. Setting x_3 = 1 from the second row we have x_2 + 2x_3 = x_2 + 2 = 0 or x_2 = -2. From the first row we then have x_1 + 2x_3 = x_1 + 2 = 0 or x_1 = -2. The vector (-2, -2, 1) is therefore a solution to Ax = 0 and a basis for the nullspace.

Since (-2, -2, 1) is in the nullspace so is its negative, (2, 2, -1), which is an alternative basis for the nullspace. Note that the inner product of u = (2, 2, -1) with v = (1, 0, 2) is

u^Tv = 2 \cdot 1 + 2 \cdot 0 + (-1) \cdot 2 = 2 - 2 = 0

and that the inner product of u = (2, 2, -1) with w = (1, 1, 4) is

u^Tw = 2 \cdot 1 + 2 \cdot 1 + (-1) \cdot 4 = 2 + 2 - 4 = 0

The basis of the nullspace u = (2, 2, -1) is thus orthogonal to each of the basis vectors for the row space, v = (1, 0, 2) and w = (1, 1, 4). The row space as a whole consists of all linear combinations c_1v+c_2w of those basis vectors, where c_1 and c_2 are scalars.

We then have

u^T(c_1v+c_2w) = c_1u^Tv + c_2u^Tw = c_1 \cdot 0 + c_2 \cdot 0 = 0

so that the basis u = (2, 2, -1) of the nullspace is orthogonal to all vectors in the row space.

Note that we have

x = (3, 3, 3) = (2, 2, -1) + (1, 1, 4) = u + w

Since the vector u (2, 2, -1) is in the nullspace and the vector w = (1, 1, 4) is in the row space (being the second row of A) we have x = x_n + x_r where x_n = u = (2, 2, -1) and x_r = w = (1, 1, 4).

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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