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karush

Gold Member

MHB

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nmh{796}

$\textsf{Suppose $Y_1$ and $Y_2$ form a basis for a 2-dimensional vector space $V$ .}\\$

$\textsf{Show that the vectors $Y_1+Y_2$ and $Y_1−Y_2$ are also a basis for $V$.}$

$$Y_1=\begin{bmatrix}a\\b\end{bmatrix}

\textit{ and }Y_2=\begin{bmatrix}c\\d\end{bmatrix}$$

$\textit{ then }$

$$Y_1+Y_2 =

\begin{bmatrix}a\\b\end{bmatrix}

-\begin{bmatrix}c\\d\end{bmatrix}

=\begin{bmatrix}a+c\\b+d\end{bmatrix}$$

$$Y_1-Y_2 =

\begin{bmatrix}a\\b\end{bmatrix}

-\begin{bmatrix}c\\d\end{bmatrix}

=\begin{bmatrix}a-c\\b-d\end{bmatrix}

$$

so far

$\textsf{Suppose $Y_1$ and $Y_2$ form a basis for a 2-dimensional vector space $V$ .}\\$

$\textsf{Show that the vectors $Y_1+Y_2$ and $Y_1−Y_2$ are also a basis for $V$.}$

$$Y_1=\begin{bmatrix}a\\b\end{bmatrix}

\textit{ and }Y_2=\begin{bmatrix}c\\d\end{bmatrix}$$

$\textit{ then }$

$$Y_1+Y_2 =

\begin{bmatrix}a\\b\end{bmatrix}

-\begin{bmatrix}c\\d\end{bmatrix}

=\begin{bmatrix}a+c\\b+d\end{bmatrix}$$

$$Y_1-Y_2 =

\begin{bmatrix}a\\b\end{bmatrix}

-\begin{bmatrix}c\\d\end{bmatrix}

=\begin{bmatrix}a-c\\b-d\end{bmatrix}

$$

so far

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