Column Picture

The column picture visualizes the system as a linear combination of column vectors.

For the system:

$$\begin{cases} 2x + y = 5 \\ x + 3y = 6 \end{cases}$$

We write this as:

$$x\begin{pmatrix}2\\1\end{pmatrix} + y\begin{pmatrix}1\\3\end{pmatrix} = \begin{pmatrix}5\\6\end{pmatrix}$$

Key Idea

Can we combine the columns with the right weights $x$ and $y$ to reach the right-hand side vector?

• Yes → unique solution (columns are independent)
• Many ways → infinitely many solutions (columns are dependent)
• No way → no solution (RHS not in column space)

Why It Matters

The column picture leads to the fundamental concept of column space — the set of all vectors reachable by linear combinations of the columns.