Consider the operation $N: P_2 \rightarrow P_3$ that takes $p(x)$ in $P_n$ to $(x-1) p(x)$ in $P_{n+1}$. Do you think this corresponds to a linear transformation from $R^3$ to $R^4$ ? If so, what is its matrix?

Prove that the range of the linear operator defined by the equations

$$\begin{aligned} &w_1=4 x_1-2 x_2 \\ &w_2=2 x_1-x_2 \end{aligned}$$

is not all of $R^2$, and find a vector that is not in the range.

Do you think the transformation of $a x^2+b x+c$ in $P_2$ to $a$ in $P_0$ corresponds to a linear transformation from $R^3$ to $R$ ?

Consider the transformation of $a x^2+b x+c$ in $P_2$ to $|a|$ in $P_0$. Prove that it does not correspond to a linear transformation by showing that there is no matrix that maps $(a, b, c)$ in $R^3$ to $|a|$ in $R$.

Check whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. If $\|k \mathbf{u}\|=k\|\mathbf{u}\|$, then $k \geq 0$.

Check whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. If $\|\mathbf{u}-\mathbf{v}\|=0$, then $\mathbf{u}=\mathbf{v}$.

Check whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. If $\mathbf{u}$ is orthogonal to $\mathbf{v}+\mathbf{w}$, then $\mathbf{u}$ is orthogonal to $\mathbf{v}$ and $\mathbf{w}$.

Check whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. If $\mathbf{u}$ is orthogonal to $\mathbf{v}$ and $\mathbf{w}$, then $\mathbf{u}$ is orthogonal to $\mathbf{v}+\mathbf{w}$.

Check whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. If $\|\mathbf{u}+\mathbf{v}\|^2=\|\mathbf{u}\|^2+\|\mathbf{v}\|^2$, then $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.

In the accompanying figure the vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{u}-\mathbf{v}$ form a triangle in $R^2$, and $A$ denotes the angle between $\mathbf{u}$ and $\mathbf{v}$. It follows from the law of cosines in trigonometry that

$$\|\mathbf{u}-\mathbf{v}\|^2=\|\mathbf{u}\|^2+\|\mathbf{v}\|^2-2\|\mathbf{u}\|\|\mathbf{v}\| \cos \theta$$

Do you think that this formula still holds if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $R^{n !}$ ? Justify your answer.

Draw a picture to illustrate this result.

If $\mathbf{u}$ and $\mathbf{v}$ are orthogonal vectors in $R^n$ such that $\|\mathbf{u}\|=1$ and $\|\mathbf{v}\|=1$, then $d(\mathbf{u}, \mathbf{v})=$___

Let $\mathbf{x}=\mathbf{x}_0+t \mathbf{v}$ be a line in $R^n$, and let $T: R^n \rightarrow R^n$ be a linear operator on $R^n$. What kind of geometric object is the image of this line under the operator $T$ ? Discuss.

The result in part (a) states a theorem about parallelograms in $R^2$. What is the theorem?