4.7 Exercises

20 points · 4 extra credit

Change of Basis

1. 1. (3 pts) Let $B = \{\mathbf{b}_1, \mathbf{b}_2\}$ and $C = \{\mathbf{c}_1, \mathbf{c}_2\}$ be bases for $\mathbb{R}^2$, and suppose $\mathbf{b}_1 = 4\mathbf{c}_1 - \mathbf{c}_2$ and $\mathbf{b}_2 = -6\mathbf{c}_1 + \mathbf{c}_2$. Write the change-of-coordinates matrix $\underset{C\leftarrow B}{P}$, and use it to find $[\mathbf{x}]_C$ when $[\mathbf{x}]_B = (3, -2)$.

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2. 2. (3 pts) Let $B = \{(1,2),\,(1,3)\}$ and $C = \{(1,1),\,(0,1)\}$ be bases for $\mathbb{R}^2$. Compute $\underset{C\leftarrow B}{P} = P_C^{-1}P_B$.

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3. 3. (3 pts) In $P_2$, let $B = \{1, t, t^2\}$ and $C = \{1,\; 1+t,\; 1+t+t^2\}$. Find $\underset{C\leftarrow B}{P}$, and use it to convert $[p]_B = (2, 3, -1)$ to $[p]_C$.

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4. 4. (3 pts) Using the matrix $\underset{C\leftarrow B}{P}$ from Problem 3, find $\underset{B\leftarrow C}{P}$, and verify that the two matrices are inverses.

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5. 5. (2 pts) Explain why the change-of-coordinates matrix $\underset{C\leftarrow B}{P}$ between two bases of $\mathbb{R}^n$ is always invertible, and identify its inverse.

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6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) The columns of $\underset{C\leftarrow B}{P}$ are the $C$-coordinate vectors of the vectors in $B$.

   (b) $\underset{C\leftarrow B}{P}\ \underset{B\leftarrow C}{P} = I$.

   (c) If $B = C$, then $\underset{C\leftarrow B}{P} = I$.

   (d) For the standard basis $E$ of $\mathbb{R}^n$, $\underset{E\leftarrow B}{P}$ is the matrix whose columns are the vectors of $B$.

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7. 7. (3 pts) Let $B$ and $C$ be bases for $\mathbb{R}^n$. Prove that $[\mathbf{x}]_C = \underset{C\leftarrow B}{P}\,[\mathbf{x}]_B$ for all $\mathbf{x}$, where the $j$-th column of $\underset{C\leftarrow B}{P}$ is $[\mathbf{b}_j]_C$.

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8. 8. (4 pts ••) Let $B$, $C$, and $D$ be bases for $\mathbb{R}^n$. Prove that $\underset{D\leftarrow B}{P} = \underset{D\leftarrow C}{P}\ \underset{C\leftarrow B}{P}$.

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