How do I give an algebraic specification for the range of a matrix?

Question

I am given the following $3 \times 3$ matrix:

$$\begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 1 \\ 2 & 2 & 10 \end{bmatrix} .$$

Once reduced, I get the following augmented matrix: $$\left[ \begin{array}{ccc|c} 1 & 0 & 7 & a-2(b-a) \\ 0 & 1 & -2 & b-a \\ 0 & 0 & 0 & c-2(b-a) \end{array} \right].$$

I don't know how to get an equation with this. I don't have 3 linearly-independent columns, I only have

$$x_1 + 7x_3 = 3a-2b \\ x_2 - 2x_3 = b-a \\ c- 2b + 2a = 0 .$$

I don't understand, am I supposed to use $x_3 = c - 2b + 2a = 0$ and plug this into the rest of the equations because I am not getting the answers in the back of the book. My book sucks at explaining.

Answer 1

Hint:what is the range of a matrix? So, if you have a matrix, you've also a linear map and the subspace spanned by the columns of the matrix is...

Answer 2

If the range is the image space of the matrix (that is the span of its column vectors), then if you only have 2 leading ones in its reduced form, what does that tell you about how many linearly independent column vectors of the matrix? Furthermore, what does that tell you about how many vectors you need to span the $col(A)$? (some of them are linear combinations of the other ones (which other ones?))

To fully answer your question, you don't need 3 column vectors that are linearly independent to span a $3\ x \ 3$ matrix. If you only have two leading ones, that already tells you the dimension of the column space. If your analysis is correct (your calculations), then the column vectors: $$\begin{bmatrix} 1 \\ 2 \\ 3\end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ 1\end{bmatrix} .$$ span the column space.