Lecture Notes

1. True or False

• Each vector space comes equipped with a preferred inner product.
• Each vector spaces comes equipped with a preferred basis.
• If $v \in V$ is an element in a vector space $V$, then it determines a dual element $v^* \in V^*$.
• Let $P$ be the vector space of smooth $\R$-valued function on $[0,1]$. For example, $f(x) = x^2-2$, is an element in $P$, or $f(x) = \frac{1}{x+1}$. Then, the function $\Phi: P \to \R$, defined by sending $f(x) \in P$ to $\int_0^1 x^2 f(x) dx$ is a linear function on $P$.

2. Area

• Consider the vector space $\R^2$. Let $\vec v_1 = (1,1), \vec v_2 = (0,2)$. Let $P(\vec v_1, \vec v_2)$ denote the area of the parallelogram (skewed rectangle) generated by $\vec v_1, \vec v_2$. $$ P(\vec v_1, \vec v_2) = ? $$
• From the above computation, can you deduce $$ P(\vec v_1 + 3 \vec v_2, \vec v_2) = ?$$ which formula did you use?
• How about $$ P(a \vec v_1 + b \vec v_2, c \vec v_1 + d \vec v_2)=? $$

3. Metric tensor. Let $V$ be a 2-dim vector space with metric tensor $g$. Let $e_1, e_2$ be a basis of $V$. Suppose we know that $$ g(e_1, e_1) = 3, g(e_1, e_2) = 1, g(e_2, e_2) = 2 $$ Answer the following question, recall that $\| v \|^2 = g(v,v)$.

• $\| e_1 \| = ? $, $\| e_2 \| = ? $
• $ \| e_1 + e_2 \| = ? $
• $ \| e_1 - 2 e_2 \| = ? $
• $P(e_1, e_2) = $?
• Can you find two vectors $v_1, v_2 \in \R^2$, such that $v_1, v_2$ has the same properties as $e_1, e_2$?

4. Let $V=\R^2$ be the Euclidean vector space of 2-dim, and $v, w$ be two vectors in it. Suppose we know that $$ g(v,v) = 1, \quad g(w,w) = 4, \quad g(v,w) = 2 $$ Can you deduce that $v$ and $w$ are collinear? (i.e. parallel? )

• What if $V$ is $n$-dimensional, does the above conclusion still holds?

1. Tangent vector of a subspace in $\R^2$. Let $S^1$ denote the unit circle in $\R^2$, i.e $$ S^1 = \{(x,y) \in \R^2 \mid x^2 + y^2 = 1 \}. $$ Let $v = (0,1)$, then for which point $p \in S^1$, is the vector $(p,v)$ a tangent vector of $S^1$ at $p$?

2. Let $f(x,y) = x^2 - y^2$, and let $\Gamma_f = \{(x,y, z) \mid z = f(x,y) \}$ the graph of $f$ in $\R^3$. Then for the point $p=(2, 3, -5)$ on $\Gamma_f$, find two linearly independent tangent vectors in $T_p \Gamma_f$.