031 Review Part 2, Problem 2

Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?

${\begin{bmatrix}1\\0\\2\end{bmatrix}},{\begin{bmatrix}3\\1\\1\end{bmatrix}},{\begin{bmatrix}-2\\-1\\1\end{bmatrix}},{\begin{bmatrix}5\\2\\2\end{bmatrix}}$

Foundations:
1.  ${\text{dim Col }}A$  is the number of pivots in  $A.$
2. A set of vectors  $\{{\vec {v_{1}}},{\vec {v_{2}}},\ldots ,{\vec {v_{n}}}\}$  is linearly independent if
the only solution to  $x_{1}{\vec {v_{1}}}+x_{2}{\vec {v_{2}}}+\cdots +x_{n}{\vec {v_{n}}}={\vec {0}}$  is the trivial solution.

Solution:

Step 1:

Step 2:

Final Answer:

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