Matrices as Linear Maps

A matrix is more than a table of numbers — it is a precise rule for transforming one vector into another. That rule is called a linear map, and it sits at the heart of every quantum gate, observable, and evolution operator you will encounter.

Vectors as columns, matrices as grids

A $2 \times 2$ matrix $M$ has four entries arranged in two rows and two columns:

M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.

A column vector $\mathbf{v}$ with two entries is written

\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}.

Applying $M$ to $\mathbf{v}$ — written $M\mathbf{v}$ — produces a new column vector whose entries are computed by taking dot products of each row of $M$ with the column $\mathbf{v}$ :

M\mathbf{v} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} a\,v_1 + b\,v_2 \\ c\,v_1 + d\,v_2 \end{pmatrix}.

The first output component is the dot product of row 0 with $\mathbf{v}$ ; the second is the dot product of row 1 with $\mathbf{v}$ .

Why "linear map"?

The map $\mathbf{v} \mapsto M\mathbf{v}$ is linear because it preserves two operations:

Addition: $M(\mathbf{u} + \mathbf{v}) = M\mathbf{u} + M\mathbf{v}$ .
Scalar multiplication: $M(c\,\mathbf{v}) = c\,(M\mathbf{v})$ for any scalar $c$ .

Both follow directly from the row-dot-column definition above. Linearity is not a coincidence — it is the property behind the superposition principle, which makes quantum evolution predictable. (Reversibility is a separate, stronger requirement: it comes from the gate being unitary, not from linearity alone.)

A concrete example

Let

M = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \qquad \mathbf{v} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}.

Applying the rule:

M\mathbf{v} = \begin{pmatrix} 1 \cdot 1 + 2 \cdot 0 \\ 3 \cdot 1 + 4 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}.

Notice what happened geometrically: $\mathbf{v} = |0\rangle = (1,0)^T$ is the first standard basis vector, and $M\mathbf{v}$ is exactly the first column of $M$ . This is a general fact: applying any matrix to a basis vector simply reads off the corresponding column.

Matrices as quantum gates

Every single-qubit quantum gate is a $2 \times 2$ matrix acting on the two-component state vector $(\alpha, \beta)^T$ . For example, the Pauli-X gate

X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

swaps the two components: $X|0\rangle = |1\rangle$ and $X|1\rangle = |0\rangle$ — a quantum bit-flip. You can verify this by applying the matrix-times-column recipe to each basis vector.

Try it

This is a numerical exercise — your code should return a number. Apply the matrix $M = \bigl(\begin{smallmatrix}1 & 2\\ 3 & 4\end{smallmatrix}\bigr)$ to the vector $\mathbf{v} = (1,\,0)^T$ and return the first component of the result.

Run your code to see the quantum state.