728862…: “Between a funny theorem”

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True/False: Any set that is a basis for a vector space $V$ is also a linearly independent set.

True False

True/False: Any linearly independent set that is a subset of $V$ is a basis for $V$ .

True False

True/False: If the $\Span \{\vec {b}_1,\dots ,\vec {b}_p\}$ is equal to some vector space $V$ , then $\{\vec {b}_1,\dots ,\vec {b}_p\}$ is a basis for $V$ .

True False

True/False: The pivot columns of $A$ form a basis for $\Col A$ .

True False

True/False: Suppose the matrix $A$ below is row equivalent to the matrices $U_1$ and $U_2$ below. Which of the following statements is true? $A= \begin {bmatrix}2 & 6& 4 &7\\ -2& 3&-4&2\\-6&0&-12&-8\end {bmatrix} \hspace {20pt} U_1 = \begin {bmatrix}2 & 0 & 4 &3\\ 0& 3&0&2\\0&0&0&1\end {bmatrix} \hspace {20pt}U_2= \begin {bmatrix}1 & 0 & 2 &0\\ 0& 1&0&0\\0&0&0&1\end {bmatrix}$

$\left \{ \begin {bmatrix} 2\\0\\0 \end {bmatrix}, \begin {bmatrix} 0\\3\\0 \end {bmatrix}, \begin {bmatrix} 3\\2\\1 \end {bmatrix}\right \}$ forms a basis for $\Col A$ . $\left \{ \begin {bmatrix} 1\\0\\0 \end {bmatrix}, \begin {bmatrix} 0\\1\\0 \end {bmatrix}, \begin {bmatrix} 0\\0\\1 \end {bmatrix}\right \}$ forms a basis for $\Col A$ . $\left \{ \begin {bmatrix} 2\\-2\\-6 \end {bmatrix}, \begin {bmatrix} 6\\3\\0 \end {bmatrix}, \begin {bmatrix} 7\\2\\-8 \end {bmatrix}\right \}$ forms a basis for $\Col A$ .