031 Review Part 2, Problem 2

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:
We begin by putting these vectors together in a matrix. So, we have
${\begin{bmatrix}1&3&-2&5\\0&1&-1&2\\2&1&1&2\end{bmatrix}}.$
Now, we row reduce this matrix. We get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\left[\begin{array}{cccc} 1 & 3 & -2 & 5 \\ 0 & 1 & -1 & 2 \\ 2 & 1 & 1 & 2 \end{array}\right]} & \sim & \displaystyle{\left[\begin{array}{cccc} 1 & 3 & -2 & 5 \\ 0 & 1 & -1 & 2 \\ 0 & -5 & 5 & -8 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{cccc} 1 & 3 & -2 & 5 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 2 \end{array}\right]} \end{array}}

Step 2:
Now, we have 3 pivots in this matrix. So, the dimension of the column space of the matrix we started with is 3.
Hence, the dimension of the subspace spanned by these vectors is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3.}
When we row reduced the matrix, we had a column that did not contain a pivot.
This means we have a free variable in the system corresponding to  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ax=0.}
So, these vectors are not linearly independent.

Final Answer:
The dimension is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}   and the vectors are not linearly independent.

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