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

Linear Algebra Problems and Solutions

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.

 
LoadingAdd to solve later

Sponsored Links

Solution.

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.


LoadingAdd to solve later

Sponsored Links

More from my site

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Problems and solutions in Linear Algebra
Subset of Vectors Perpendicular to Two Vectors is a Subspace

Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by \[W=\{\mathbf{x}\in...

Close