If Vectors are Linearly Dependent, then What Happens When We Add One More Vectors?

Problem 120

Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$ are linearly dependent $n$-dimensional real vectors.

For any vector $\mathbf{v}_{r+1} \in \R^n$, determine whether the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly independent or linearly dependent.

We claim that the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly dependent.
Since the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$ are linearly dependent, there exist scalars (real numbers) $a_1, a_2, \dots, a_r$ such that
\[a_1 \mathbf{v}_1+a_2\mathbf{v}_2+\cdots +a_r\mathbf{v}_r=\mathbf{0} \tag{*}\]
and not all of $a_1, \dots, a_r$ are zero, that is, $(a_1, \dots, a_r) \neq (0, \dots, 0)$.

Consider the equation
\[x_1\mathbf{v}_1+x_2 \mathbf{v}_2+\cdots +x_r \mathbf{v}_r+x_{r+1} \mathbf{v}_{r+1}=\mathbf{0}.\]
If this equation has a nonzero solution $(x_1, \dots, x_r, x_{r+1})$, then the vectors $\mathbf{v}_1, \dots, \mathbf{v}_{r+1}$ are linearly dependent.

In fact,
\[(x_1,x_2,\dots, x_r, x_{r+1})=(a_1, a_2, \dots, a_r, 0)\]
is a nonzero solution of the above equation.
To see this, first note that since not all of $a_1, a_2, \dots, a_r$ are zero, we have
\[(a_1, a_2, \dots, a_r, 0)\neq (0, 0, \dots, 0, 0).\]

Plug these values in the equation, we have
\begin{align*}
&a_1 \mathbf{v}_1+a_2\mathbf{v}_2+\cdots +a_r\mathbf{v}_r+0\mathbf{v}_{r+1}\\
&=a_1 \mathbf{v}_1+a_2\mathbf{v}_2+\cdots +a_r\mathbf{v}_r=\mathbf{0} \text{ by (*).}
\end{align*}
Therefore, we conclude that the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly dependent.

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