Question

The following statement is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If the statement is true, give a justification. If $v_1, v_2, v_3$ are in $R^3$ and $v_3$ is not a linear combination of $v_1, v_2$, then $(v_1, v_2, v_3)$ is linearly independent. Fill in the blanks below. The statement is false. Take $v_1$ and $v_2$ to be multiples of one vector and take $v_3$ to be not a multiple of that vector. For example, $\begin{bmatrix} 1\\1\\1 \end{bmatrix} = v_1$, $\begin{bmatrix} 2\\2\\2 \end{bmatrix} = v_2$, $\begin{bmatrix} 1\\0\\0 \end{bmatrix} = v_3$. Since at least one of the vectors is a linear combination of the other two, the three vectors are linearly dependent independent

          The following statement is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If the statement is true, give a justification.
If $v_1, v_2, v_3$ are in $R^3$ and $v_3$ is not a linear combination of $v_1, v_2$, then $(v_1, v_2, v_3)$ is linearly independent.
Fill in the blanks below.
The statement is false. Take $v_1$ and $v_2$ to be multiples of one vector and take $v_3$ to be not a multiple of that vector. For example,
$\begin{bmatrix} 1\\1\\1 \end{bmatrix} = v_1$, $\begin{bmatrix} 2\\2\\2 \end{bmatrix} = v_2$, $\begin{bmatrix} 1\\0\\0 \end{bmatrix} = v_3$. Since at least one of the vectors is a linear combination of the other two, the three vectors are linearly
dependent
independent

The following statement is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If the statement is true, give a justification.
If v1, v2, v3 are in R^3 and v3 is not a linear combination of v1, v2, then (v1, v2, v3) is linearly independent.
Fill in the blanks below.
The statement is false. Take v1 and v2 to be multiples of one vector and take v3 to be not a multiple of that vector. For example,
< b m a t r i x >
= v1, < b m a t r i x >
= v2, < b m a t r i x >
= v3. Since at least one of the vectors is a linear combination of the other two, the three vectors are linearly
dependent
independent

Added by Cole P.

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