To determine whether the given vectors are linearly dependent, we'll go through a series of steps to analyze them. Here are the given vectors:
[tex]\[
a_1 = \begin{pmatrix} 2 \\ 16 \end{pmatrix}, \quad a_2 = \begin{pmatrix} 4 \\ 7 \end{pmatrix}, \quad a_3 = \begin{pmatrix} 8 \\ 10 \end{pmatrix}
\][/tex]
### Step 1: Forming the Matrix
We start by forming a matrix [tex]\( A \)[/tex] whose columns are the given vectors:
[tex]\[
A = \begin{pmatrix}
2 & 4 & 8 \\
16 & 7 & 10
\end{pmatrix}
\][/tex]
### Step 2: Determine the Rank of the Matrix
The rank of a matrix is the maximum number of linearly independent column vectors in the matrix. To check this, we need to determine the rank of matrix [tex]\( A \)[/tex].
### Step 3: Checking Linearly Dependent Vectors
In general, if the number of vectors [tex]\( n_v \)[/tex] is greater than the rank [tex]\( r \)[/tex] of the matrix formed by these vectors, then the vectors are linearly dependent.
- The rank of matrix [tex]\( A \)[/tex] is 2.
- The number of vectors, which corresponds to the number of columns in matrix [tex]\( A \)[/tex], is 3.
Since the rank of the matrix [tex]\( A \)[/tex] (which is 2) is less than the number of vectors (which is 3), we conclude that the vectors are linearly dependent.
### Conclusion
Given the vectors [tex]\( a_1, a_2, \)[/tex] and [tex]\( a_3 \)[/tex], we determined that they are indeed linearly dependent.
