0b9715…: “Out of the beautiful parallelogram”

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True/False: The standard basis for $\RR ^2$ is $\mathcal {B} = \left \{\begin {bmatrix}0\\1\end {bmatrix},\begin {bmatrix}1\\0\end {bmatrix}\right \}$ .

True False

True/False: Let $\vec {x}$ be in some vector space $V$ and let $\mathcal {B} = \{\vec {b}_1, \dots ,\vec {b}_n\}$ be a basis for $V$ . Then $\vec {x}$ can be written in two different ways: $\vec {x}= c_1\vec {b}_1+\cdots +c_n\vec {b}_n \hspace {10pt} \text {and }\hspace {10pt} \vec {x}= d_1\vec {b}_1+\cdots +d_n\vec {b}_n$ where not all of the $c_i$ ’s are equal to the corresponding $d_i$ ’s.

True False

Check out the unique representation theorem.

Suppose $\mathcal {B} = \{\vec {b}_1, \vec {b}_2, \vec {b}_3\}$ is a basis for some vector space $V$ and $\vec {x} = 3\vec {b}_1 -2\vec {b}_2 +8\vec {b}_3$ is a vector in $V$ . What is $[\vec {x}]_{\mathcal {B}}$ ?

$[\vec {x}]_{\mathcal {B}} = \begin {bmatrix} \answer {3}\\ \answer {-2} \\\answer {8} \end {bmatrix}$

$\mathcal {B} =\left \{\begin {bmatrix}4\\2\end {bmatrix}, \begin {bmatrix} -2\\2\end {bmatrix}\right \}$ is a basis for $\RR ^2$ and $\vec {x} = \begin {bmatrix} 8\\10\end {bmatrix}$ is a vector in $\RR ^2$ . Find the coordinate vector of $\vec {x}$ relative to $\mathcal {B}$

$[\vec {x}]_{\mathcal {B}} = \begin {bmatrix} \answer {3}\\\answer {2} \end {bmatrix}$

$\mathcal {B} =\left \{\begin {bmatrix}2\\3\end {bmatrix}, \begin {bmatrix} -1\\6\end {bmatrix}\right \}$ is a basis for $\RR ^2$ and $\vec {x} = \begin {bmatrix} 5\\1\end {bmatrix}$ is a vector in $\RR ^2$ . Find the coordinate vector of $\vec {x}$ relative to $\mathcal {B}$

$[\vec {x}]_{\mathcal {B}} = \begin {bmatrix} \frac {\answer {31}}{\answer {15}}\\ -\frac {\answer {13}}{\answer {15}} \end {bmatrix}$

Let $\mathcal {B} = \left \{ \begin {bmatrix} 1\\0\\3\end {bmatrix}, \begin {bmatrix} 2\\-1\\0\end {bmatrix}, \begin {bmatrix} 1\\1\\-2\end {bmatrix} \right \}$ be a basis for $\RR ^3$ . If $[\vec {x}]_{\mathcal {B}} = \begin {bmatrix} 1\\3\\5\end {bmatrix}$ , what is $\vec {x}$ , that is the same vector in $\RR ^3$ , but written in terms of the standard basis of $\RR ^3$ ?

$\vec {x} = \begin {bmatrix} \answer {12}\\\answer {2}\\ \answer {-7}\end {bmatrix}$

$\vec {x} = P_{\mathcal {B}} [\vec {x}]_{\mathcal {B}}$

Let $\mathcal {B}$ be a basis for $\RR ^2$ . If $\vec {x} = \begin {bmatrix} -17\\ 5\end {bmatrix}$ and $[\vec {x}]_{\mathcal {B}} = \begin {bmatrix} 3\\ -4\end {bmatrix}$ , then which of the following is the basis $\mathcal {B}$ ?

$\left \{ \begin {bmatrix} 1\\2\end {bmatrix}, \begin {bmatrix} 0\\2\end {bmatrix} \right \}$ $\left \{ \begin {bmatrix} -17\\5\end {bmatrix}, \begin {bmatrix} 3\\-4\end {bmatrix} \right \}$ $\left \{ \begin {bmatrix} 1\\3\end {bmatrix}, \begin {bmatrix} 5\\1\end {bmatrix} \right \}$

$\vec {x} = P_{\mathcal {B}} [\vec {x}]_{\mathcal {B}}$