Linear Algebra

The Length of a Vector is Zero if and only if the Vector is the Zero Vector

Problem 639

Let $\mathbf{v}$ be an $n \times 1$ column vector.

Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$.

Proof.

Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} $.

Then we have
\[\mathbf{v}^\trans \mathbf{v} = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} = \sum_{i=1}^n \mathbf{v}_i^2 . \] Because each $v_i^2$ is non-negative, this sum is $0$ if and only if $v_i = 0$ for each $i$. In this case, $\mathbf{v}$ is the zero vector.

Comment.

Recall that the the length of the vector $\mathbf{v}\in \R^n$ is defined to be
\[\|\mathbf{v}\| :=\sqrt{\mathbf{v}^{\trans} \mathbf{v}}.\]

The problem implies that the length of a vector is $0$ if and only if the vector is the zero vector.

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