4.4 Exercises

20 points · 4 extra credit

Coordinate Systems

1. 1. (2 pts) Let $B = \{(1,2),\,(3,5)\}$, a basis for $\mathbb{R}^2$. Find the coordinate vector $[\mathbf{x}]_B$ for $\mathbf{x} = (4, 5)$.

   #

2. 2. (2 pts) Let $B = \{(2,1),\,(-1,1)\}$, a basis for $\mathbb{R}^2$. Given $[\mathbf{x}]_B = (3, 2)$, find $\mathbf{x}$.

   #

3. 3. (3 pts) Let $B = \{1,\; 1+t,\; 1+t+t^2\}$, a basis for $P_2$. Find $[p]_B$ for $p(t) = 2 + 3t - t^2$.

   #

4. 4. (3 pts) Let $B = \{(1,0,1),\,(2,1,0),\,(0,1,1)\}$, a basis for $\mathbb{R}^3$. Write down the change-of-coordinates matrix $P_B$, and use it to find $\mathbf{x}$ when $[\mathbf{x}]_B = (1, -1, 2)$.

   #

5. 5. (3 pts) Let $B = \{(2,1),\,(1,1)\}$, a basis for $\mathbb{R}^2$. Compute $P_B^{-1}$ and use it to find $[\mathbf{x}]_B$ for $\mathbf{x} = (3, 2)$.

   #

6. 6. (2 pts) State whether each of the following is true or false, and justify each answer.

   (a) The coordinate vector $[\mathbf{x}]_B$ depends on the order in which the basis vectors are listed.

   (b) If $B$ is a basis for $\mathbb{R}^n$, then the change-of-coordinates matrix $P_B$ is invertible.

   (c) For the standard basis $E$ of $\mathbb{R}^n$, $[\mathbf{x}]_E = \mathbf{x}$.

   #

7. 7. (3 pts) Let $B = \{\mathbf{b}_1, \ldots, \mathbf{b}_n\}$ be a basis for a vector space $V$. Prove that the coordinate mapping $\mathbf{x}\mapsto[\mathbf{x}]_B$ is one-to-one: if $[\mathbf{x}]_B = [\mathbf{y}]_B$, then $\mathbf{x} = \mathbf{y}$.

   #

8. 8. (2 pts) Do $\mathbf{b}_1 = (1,3)$ and $\mathbf{b}_2 = (2,6)$ form a basis for $\mathbb{R}^2$? If not, explain why coordinates relative to $\{\mathbf{b}_1, \mathbf{b}_2\}$ are not well defined.

   #

9. 9. (4 pts ••) Let $B = \{\mathbf{b}_1, \ldots, \mathbf{b}_n\}$ be a basis for $\mathbb{R}^n$. Prove that the coordinate mapping $\mathbf{x}\mapsto[\mathbf{x}]_B$ is a linear transformation and is onto $\mathbb{R}^n$.

   #