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Let $$B = \{ \vec{b}_1, \vec{b}_2 \}$$ and $$C = \{ \vec{c}_…

Let $$B = \{ \vec{b}_1, \vec{b}_2 \}$$ and $$C = \{ \vec{c}_1, \vec{c}_2 \}$$ be bases for a vector space V, and suppose $$\vec{b}_1 = \begin{bmatrix}&2\\&3\end{bmatrix}$$ , $$\vec{b}_2 = \begin{bmatrix}&6\\&7\end{bmatrix} $$, $$\vec{c}_1 =\begin{bmatrix}&2\\&5\end{bmatrix}$$, and $$\vec{c}_2 = \begin{bmatrix}&4\\&2\end{bmatrix}$$.   a) Find the change-of-coordinate matrix from $B$ to $C$.   b) Using part a) Find the change-of-coordinate matrix from C to B.   c) Let $$[\vec{x}]_{C} = \begin{bmatrix}&1\\&\frac{1}{2}\end{bmatrix}$$. Find $$[\vec{x}]_{B}$$.  

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The two parts of this problem are independent.   a) Show tha…

The two parts of this problem are independent.   a) Show that if $$||\vec{u}-\vec{v}||^2 = ||\vec{u}+\vec{v}||^2$$ then $$\vec{u}$$ and $$\vec{v}$$ are orthogonal.   b) Let $$\{\vec{u}_1, \vec{u}_2, \vec{u}_3, \vec{u}_4\}$$ be an orthogonal basis for $$R^4$$. Let W be Span $$\{\vec{u}_1, \vec{u}_2, \vec{u}_3\}$$. Write $$\vec{x}$$ as the sum of two vectors, one in W and the other perpendicular to W. $$\vec{u}_1 = \begin{bmatrix}&1 \\&1 \\&0 \\&-1\end{bmatrix}$$, $$\vec{u}_2 = \begin{bmatrix}&1 \\&0 \\&1 \\&1\end{bmatrix}$$, $$\vec{u}_3 = \begin{bmatrix}&0 \\&-1 \\&1 \\&-1\end{bmatrix}$$, and $$\vec{x} = \begin{bmatrix}&-2 \\&3 \\&6 \\&-4\end{bmatrix}$$  

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