Answer
See explanation
Work Step by Step
\[
A=\left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
2 & 5 & 0 & 0 \\
3 & 6 & 8 & 0 \\
4 & 7 & 9 & 10
\end{array}\right]
\]
Goal:
Knowing why the columns of matrix A are linearly independent using Theorem 7
Concepts:
Linear Independence Theorem 7
1
Solve
$\mathbf{v}_{1}, \ldots \mathbf{v}_{4}$ denote the columns of $A .$ Observe the following:
$\mathbf{v}_{1}$ is non-zero. $\mathrm{v}_{2}$ is not a multiple of $\mathbf{v}_{1}$ $\mathbf{v}_{3}$ is not a linear combination of $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ $\mathbf{v}_{4}$ canot be a linear combination of $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$
Conclusion
The columns of A are linearly independent.
