Let $\mathbf{0}$ be the zero vector, and $\mathbf{v}_1, \cdots, \mathbf{v}_k$ are the other vectors in the set.

Then we have the non-trivial linear combination
\[1 \cdot \mathbf{0} + 0 \mathbf{v}_1 + 0 \mathbf{v}_2 + \cdots + 0 \mathbf{v}_k = \mathbf{0}.\]
This is a non-trivial linear combination because one of the coefficients is non-zero.

Thus by definition, the set $\{ \mathbf{0} , \mathbf{v}_1 , \cdots , \mathbf{v}_k \}$ is linearly dependent.

The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero
Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$.
Then prove that $V$ is a subspace of $\R^n$.
Proof.
To prove that $V=\{\mathbf{0}\}$ is a subspace of $\R^n$, we check the following subspace […]

Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis
Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient.
Let $W$ be the following subset of $P_3$.
\[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\]
Here $p'(x)$ is the first derivative of $p(x)$ and […]

Determine a Condition on $a, b$ so that Vectors are Linearly Dependent
Let
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
1 \\
a \\
5
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
0 \\
4 \\
b
\end{bmatrix}\]
be vectors in $\R^3$.
Determine a […]