Linear Algebra and Its Applications, Exercise 3.3.6

Exercise 3.3.6. Given the matrix $A$ and vector $b$ defined as follows

$A = \begin{bmatrix} 1&1 \\ 2&-1 \\ -2&4 \end{bmatrix} \qquad b = \begin{bmatrix} 1 \\ 2 \\ 7 \end{bmatrix}$

find the projection of $b$ onto the column space of $A$ .

Decompose the vector $b$ into the sum $p + q$ of two orthogonal vectors $p$ and $q$ where $p$ is in the column space. Which subspace is $q$ in?

Answer: We have $p = A(A^TA)^{-1}A^Tb$ per equation (3) of 3L on page 156. We first compute

$A^TA = \begin{bmatrix} 1&2&-2 \\ 1&-1&4 \end{bmatrix} \begin{bmatrix} 1&1 \\ 2&-1 \\ -2&4 \end{bmatrix}$

$= \begin{bmatrix} 9&-9 \\ -9&18 \end{bmatrix}$

and then compute

$(A^TA)^{-1} = \frac{1}{9 \cdot 18 - (-9)(-9)} \begin{bmatrix} 18&-(-9) \\ -(-9)&9 \end{bmatrix}$

$= \frac{1}{81} \begin{bmatrix} 18&9 \\ 9&9 \end{bmatrix} = \begin{bmatrix} \frac{2}{9}&\frac{1}{9} \\ \frac{1}{9}&\frac{1}{9} \end{bmatrix}$

Finally we compute

$p = A(A^TA)^{-1}A^Tb$

$= \begin{bmatrix} 1&1 \\ 2&-1 \\ -2&4 \end{bmatrix} \begin{bmatrix} \frac{2}{9}&\frac{1}{9} \\ \frac{1}{9}&\frac{1}{9} \end{bmatrix} \begin{bmatrix} 1&2&-2 \\ 1&-1&4 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ 7 \end{bmatrix}$

$= \begin{bmatrix} 1&1 \\ 2&-1 \\ -2&4 \end{bmatrix} \begin{bmatrix} \frac{2}{9}&\frac{1}{9} \\ \frac{1}{9}&\frac{1}{9} \end{bmatrix} \begin{bmatrix} -9 \\ 27 \end{bmatrix}$

$= \begin{bmatrix} 1&1 \\ 2&-1 \\ -2&4 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ 6 \end{bmatrix}$

Since $b = p + q$ we have

$q = b - p = \begin{bmatrix} 1 \\ 2 \\ 7 \end{bmatrix} - \begin{bmatrix} 3 \\ 0 \\ 6 \end{bmatrix} = \begin{bmatrix} -2 \\ 2 \\ 1 \end{bmatrix}$

We have

$p \cdot q = 3 \cdot (-2) + 0 \cdot 2 + 6 \cdot 1 = -6 + 0 + 6 = 0$

so that $p$ and $q$ are orthogonal.

The vector $p$ is in $\cal{R}(A)$ , the column space of A, and the orthogonal subspace of $\cal{R}(A)$ is $\cal{N}(A^T)$ , the left nullspace of $A$ . Since $p$ is in $\cal{R}(A)$ and $q$ is orthogonal to $p$ , $q$ must be in $\cal{N}(A^T)$ , so that $A^Tq = 0$ . We confirm this:

$A^Tq = \begin{bmatrix} 1&2&-2 \\ 1&-1&4 \end{bmatrix} \begin{bmatrix} -2 \\ 2 \\ 1 \end{bmatrix}$

$= \begin{bmatrix} -2 + 4 -2 \\ -2 - 2 + 4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} = 0$

UPDATE: I corrected the calculation of $q$ ; thanks go to KTL for pointing out the error.

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.

6 Responses to Linear Algebra and Its Applications, Exercise 3.3.6

Teresa says:

July 18, 2016 at 12:31 am

Thank you for the great work! It’s been really helpful. I wanted to mention that A transpose by A is [6 -8; -8 18] and not [9 -9; -9 18] that changes the results a bit.

hecker says:

July 18, 2016 at 2:40 am

I’m glad you find these posts useful. I’m sorry it’s been a long time since I published the last one.

I don’t understand your comment. Are you referring to A transpose multiplied by A (on the right)? If so, the (1, 1) element of the result matrix is 1 x 1 + 2 x 2 + (-2) x (-2) = 1 + 4 + 4 = 9. I don’t know how you got 6 for that result. Similarly the (1, 2) entry is 1 x 1 + 2 x (-1) + (-2) x 4 = 1 – 2 – 8 = -9 (not -8) and the (2, 1) entry is 1 x 1 + (-1) x 2 + 4 x (-2) = 1 – 2 – 8 = -9 again.

- KTL says:
  
  September 26, 2016 at 2:43 pm
  
  The question in the 4th edition is different. Hence the confusion for the person above 🙂
  
  - hecker says:
    
    September 26, 2016 at 4:50 pm
    
    Ah, thanks for the explanation. I’ve never looked at a copy of the 4th edition so I don’t know how the exercises differ.
KTL says:

September 26, 2016 at 2:42 pm

q you have given as p-b though the calculation is done as b-p (correctly)

- hecker says:
  
  September 26, 2016 at 4:45 pm
  
  Thanks for finding this error! I’ve corrected the post.