A
component-wise proof
is a proof type that is applicable to statements involving vectors. A similar idea can be applied to matrices. The basic idea is to write a statement about vectors in an expanded form where the vectors are expressed in terms of the components of those vectors. Since the components are just real numbers, we can apply things that we already know about real numbers to the situation at hand.
Example
The dot product of 2D vectors is a commutative operation.
Proof
: Let $\bf u$ and $\bf v$ be 2D vectors. Then, they may be written in the form
$${\bf u} = \langle u_1,u_2 \rangle \: \text{ and } \: {\bf v} = \langle v_1,v_2 \rangle.$$
Thus,
$$\begin{array}{cll}
{\bf u}\cdot{\bf v} & = & \langle u_1,u_2 \rangle \cdot \langle v_1,v_2 \rangle = u_1v_1 + u_2v_2 \\
& = & v_1u_1 + v_2u_2 = \langle v_1,v_2 \rangle \cdot \langle u_1,u_2 \rangle = {\bf v} \cdot {\bf u}.\Box
\end{array}$$
Note that the first step was to introduce symbols $\bf u$ and $\bf v$ to represent 2D vectors and to then write them in an expanded form
$${\bf u} = \langle u_1,u_2 \rangle \: \text{ and } \: {\bf v} = \langle v_1,v_2 \rangle.$$
That's what makes this a component-wise proof.
Comments
A component-wise proof is a proof type that is applicable to statements involving vectors. A similar idea can be applied to matrices. The basic idea is to write a statement about vectors in an expanded form where the vectors are expressed in terms of the components of those vectors. Since the components are just real numbers, we can apply things that we already know about real numbers to the situation at hand.
Example
The dot product of 2D vectors is a commutative operation.
Proof : Let $\bf u$ and $\bf v$ be 2D vectors. Then, they may be written in the form
$${\bf u} = \langle u_1,u_2 \rangle \: \text{ and } \: {\bf v} = \langle v_1,v_2 \rangle.$$
Thus,
$$\begin{array}{cll}
{\bf u}\cdot{\bf v} & = & \langle u_1,u_2 \rangle \cdot \langle v_1,v_2 \rangle = u_1v_1 + u_2v_2 \\
& = & v_1u_1 + v_2u_2 = \langle v_1,v_2 \rangle \cdot \langle u_1,u_2 \rangle = {\bf v} \cdot {\bf u}.\Box
\end{array}$$
Note that the first step was to introduce symbols $\bf u$ and $\bf v$ to represent 2D vectors and to then write them in an expanded form
$${\bf u} = \langle u_1,u_2 \rangle \: \text{ and } \: {\bf v} = \langle v_1,v_2 \rangle.$$
That's what makes this a component-wise proof.