Projecting a general solution onto the row space.

Question

I'm new to linear algebra and this homework problem has had me stumped for quite a while now. Firstly, we are given 2 matrices
$A=\pmatrix{5&-1&2\cr1&-5&-2\cr8&2&5\cr}$, $b=(2,-2,5)$
Picture Of Matrixs

1. We are then asked to calculate the general solution for $Ax-b$ (easy enough - we can a matrix that spans a plane ) My general solution is : (0.5-0.5t; 0.5 -0.5t; t)

Part 2 of the question, that has me stumped is;

2. Find the orthogonal projection of the general solution $x$ onto the row space of $A$

Part 3, that I am also confused by is

3. What is the orthogonal projection of the right-hand side vector $b$ onto the column space of $A$?

Any guidance on how to solve this problem would be greatly appreciated, thank you.

Answer 1

The orthogonal projection is a linear transformation (matrix) that satisfies $P = P^2$ so it is a projection and such that $x-Px$ is orthogonal to $Px$ so that the projection is orthogonal. If $u$ is a unit column vector then the matrix $uu^T$ with $u^T$ the transpose (row vector) will be the orthogonal projection onto the subspace spanned by $u$.

Answer 2

The question is actually testing your understanding of the four fundamental subspaces of linear algebra. To have a deeper understanding of the subspaces, you can have a look at Strang's paper The Fundamental Theorem of Linear Algebra

Reducing the matrix $[A|b]$ to its row echelon form, we get

$$\pmatrix{5 & -1 & 2 &\vert&2 \\1&-5&-2&|&-2 \\8&2&5&|&5}\overset{rref}{\thicksim} \pmatrix{1 & 0 & 1/2 &\vert&2 \\0&1&1/2&|&1/2 \\0&0&0&|&0}$$

From the rref form we know that there exists infinite solution for the equation $A\mathbf{x}=\mathbf{b}$ and the matrix $A$ is not full rank. This tells us 2 things:

1. $\mathbf{b}$ falls inside the column space $R(A)$ as there exist a solution for $A\mathbf{x}=\mathbf{b}$
2. the general solution $\mathbf{x}=\mathbf{x}_r + \mathbf{x}_n$ can be decomposed to 2 vectors with $\mathbf{x}_r$ coming from row space $R(A^T)$ and $\mathbf{x}_n$ coming from the null space $N(A)$

You have answered your first question in which $$\begin{aligned}x &=\pmatrix{1/2 \\ 1/2 \\ 0} + \pmatrix{-1/2 \\ -1/2 \\ 1}t \\ &= \mathbf{x}_r + \mathbf{x}_n\end{aligned}$$

For the second question, we will only need to extract the $\mathbf{x}_r$ from $x$, so the solution will be $(1/2, 1/2, 0)^T$

For the third question, since $b$ itself is in the column space, the solution will be $(2,-2,5)^T$