Q.Find the rank of the matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$.

Linear Combination

A linear combination is what you get when you take a few objects, scale each by a number, and add the results. It is the single most important pattern in linear algebra, because scalar multiplication and addition are the only two operations it uses.

The Idea

Given objects $A_1, A_2, \dots, A_k$ (they can be vectors, or matrices of the same order) and scalars $c_1, c_2, \dots, c_k$, the linear combination is

Each $c_i A_i$ is a scalar multiple; then you add them all. The scalars are called the coefficients or weights.

A Concrete Example

With vectors $\vec{u} = (1, 0)$ and $\vec{v} = (0, 1)$:

So the point $(3, 2)$ is a linear combination of $\vec{u}$ and $\vec{v}$ with weights $3$ and $2$. The same idea works for matrices — e.g. $2A - B$ is a linear combination of $A$ and $B$ with coefficients $2$ and $-1$.

Why It Matters

• Building everything from a few pieces. Every vector in the plane is a linear combination of $\vec{i}=(1,0)$ and $\vec{j}=(0,1)$. Such a generating set is the seed of the idea of a basis.
• Asking "can I reach this?" Deciding whether $\vec{w}$ is a linear combination of given vectors is the same as asking whether a system of linear equations has a solution.
• Dependence. If one object is a linear combination of the others, the set carries redundant information (it is linearly dependent). …