# Vector Spaces

## Vector Space

• Set $$V$$, operations $$+:V \times V \to V$$ and $$\cdot: \mathbb{R} \times V \to V$$ such that
• $$(V,+)$$ is commutative group: $$\forall x,y,z \in V$$
1. (Algebraically) closed: $$x + y \in V$$
2. Associative: $$(x + y) + z = x + (y+z)$$
3. Identity: $$\exists 0 \in V$$ such that $$x + 0 = x$$
4. Invertible: $$\exists -x \in V$$ such that $$x + (-x) = 0$$
5. Commutative: $$x + y = y + x$$
• Scalar multiplication has properties on next slide

## Vector Space

• Set $$V$$, operations $$+:V \times V \to V$$ and $$\cdot: \mathbb{R} \times V \to V$$ such that
• $$(V,+)$$ is commutative group
• Scalar multiplication is: $$\forall \alpha, \beta \in \mathbb{R}$$
1. Closed: $$\alpha x \in V$$
2. Distributive: $$\alpha(x + y) = \alpha x + \alpha y$$ and $$(\alpha + \beta) x = \alpha x + \beta x$$
3. Consistent with multiplication in $$\mathbb{R}$$: $$1x = x$$ and $$(\alpha \beta) x = \alpha (\beta x)$$

## Examples

• $$\mathbb{R}^n$$
• Spaces of sequences
• Spaces of functions

## Subspaces

• $$S \subseteq V$$ is a linear subspace if $$\forall x,y \in S$$ and $$\alpha \in \mathbb{R}$$ $\alpha x + y \in S$
• Examples

## Linear Combinations

• $$\mathbf{x}_1,..., \mathbf{x}_k \in V$$, linear combination is $c_1 \mathbf{x}_1 + ... + c_k \mathbf{x}_k$ where $$c_1, ..., c_k \in \mathbb{R}$$

• span of $$W \subseteq V$$ is set of all linear combinations of elements of $$W$$

• is a linear subspace

## Linear independence

• $$W \subseteq V$$ is linearly independent if $\sum_{j=1}^k c_j \mathbf{x}_j = 0$ implies $$c_1 = c_2 = ... = c_k = 0$$ for any $$k$$ and $$\mathbf{x}_1, ..., \mathbf{x}_k \in W$$
• Dimension of a vector space is the cardinality of the largest set of linearly independent elements
• Basis $$B \subseteq V$$ is a set of linearly independent elements that span $$V$$
• Should show dimension is well-defined and basis exists

## Any basis has the same cardinality

• Lemma: If $$A$$ and $$B$$ are bases for $$V$$, then $$|A| = |B|$$

• $$A$$ basis means for each $$\mathbf{b} \in B$$, $$\exists$$ finite $$A_b \subset A$$ and $$x_{a,b} \in \mathbb{R}$$ such that $$\mathbf{b} = \sum_{\mathbf{a}\in A_b} x_{a,b} \mathbf{a}$$
• $$B \cup (A \setminus \cup_{b \in B} A_b)$$ is linearly independent, so $$A = \cup_{b \in B} A_b$$
• $$|A| \leq |B|$$

## Basis exists

• Lemma: Let $$V$$ be a vector space, then $$\exists$$ a basis

• $$\exists L \subseteq V$$ s.t $$L$$ linearly independent
• Define $$\mathcal{P} = \{S \subseteq V: L \subseteq S \text{ and } S \text{ linearly independent}\}$$
• $$\mathcal{P}, \subseteq$$ is partially ordered
• $$\overline{C} = \cup_{S \in \mathcal{C}} S$$ is upper bound for chain $$\mathcal{C}$$
• By Zorn’s lemma, $$\exists$$ maximal $$B \in \mathcal{P}$$
• Span$$(B) = V$$

## Linear independence

• Lemma: Let $$B$$ be a basis for a vector space $$V$$. Then $$\forall \mathbf{x} \in V$$ there exists a unique $$x_1, ..., x_k \in \mathbb{R}$$ and $$b_1, ..., b_k \in B$$ such that $$\mathbf{x} = \sum_{i=1}^k x_i b_i$$
• $$\mathbb{R}^n$$ is the only finite dimensional vector space

# Normed Vector Spaces

## Normed Vector Space

• $$\Vert \cdot \Vert:V \to \mathbb{R}$$ s.t.
• $$\Vert v \Vert \geq 0$$, $$=0$$ iff $$v=0$$
• $$\Vert \alpha v \Vert = |\alpha| \Vert v \Vert$$
• $$\Vert v_1 + v_2 \Vert \leq \Vert v_1 \Vert + \Vert v_2 \Vert$$
• Gives a metric $$d(v_1,v_2) = \Vert v_1 - v_2 \Vert$$

## Normed Vector Spaces

• $$\mathbb{R}^n$$ with $$\Vert \mathbf{x} \Vert = \sqrt{\sum_{i=1}^n x_i^2}$$
• $$\ell_p =$$ infinite sequences s.t. $\Vert x\Vert_p = \left( \sum_{i=1}^\infty |x_i|^p \right)^{1/p} < \infty$
• $$\mathcal{L}_p([0,1]) = \{f:[0,1] \to \mathbb{R}\}$$ s.t. $\Vert f \Vert_p = \left(\int_0^1 |f(x)|^p dx\right)^{1/p} < \infty$

# Linear Transformations

## Linear Transformations

• $$A:V \to W$$ s.t. $$\forall v_1,v_2\in V\,,\,\alpha \in \mathbb{R}$$
• $$A(v_1 + v_2) = Av_1 + Av_2$$
• $$A(\alpha v_1 ) = \alpha A v_1$$
• Represented as matrices when $$V$$ and $$W$$ finite dimensional

## Linear Transformations

• Examples:
• Integration: $$V = \{f:[0,1] \to \mathbb{R} s.t. \int_0^1 |f(x)|dx < \infty \}$$, $$A:V \to \mathbb{R}$$ $Af = \int_0^1 a(x) f(x) dx$ with $$\sup_{x \in [0,1]} |a(x)| < \infty$$

## Linear Transformations

• Examples:
• Differentiation: $$C^k([0,1]) =$$ $$\{f:[0,1] \to \mathbb{R} \;,\; f \text{ k times cont differentiable}\}$$

• $$D:C^1([0,1]) \to C^0([0,1])$$, $(Df)(x) = \frac{df}{dx}(x)$

## Linear Transformations

• Examples:
• Conditional expectation

## Vector space of linear transformations

• $$L(V,W) =$$ { linear transformations from $$V$$ to $$W$$ }
• Addition: $$(A + B)(v) = Av + Bv$$
• Scalar multiplication: $$A(\alpha v) = \alpha Av$$

## Matrix multiplication=composition of linear transformations

• $$A : V \to W$$, $$B: X \to V$$
• $$V$$ is $$n$$ dimension with basis $$f_j$$
• $$W$$ is $$m$$ dimension with basis $$e_i$$
• $$X$$ is $$p$$ dimension with basis $$g_k$$

## Matrix multiplication=composition

\begin{aligned} A(B g_k) = & A (\sum_{j=1}^n b_{jk} f_j) = \sum_{j=1}^n b_{jk} A f_j \\ = & \sum_{j=1}^n b_{jk} \left(\sum_{i=1}^m a_{ij} e_i\right) = \sum_{i=1}^m \left(\sum_{j=1}^n a_{ij} b_{jk} \right) e_i \\ = & \begin{pmatrix} \sum_{j=1}^n a_{1j} b_{j1} & \cdots & \sum_{j=1}^n a_{1j} b_{jp} \\ \vdots & \ddots & \vdots \\ \sum_{j=1}^n a_{mj} b_{j1} & \cdots & \sum_{j=1}^n a_{mj} b_{jp} \end{pmatrix} g_k \\ = & (AB)g_k . \end{aligned}

## Null space and range

• $$A \in L(V,w)$$, null space of $$A$$ $null(A) = \{x \in V: Ax = 0\} \subseteq V$
• range of $$A$$ $range(A) = \{Ax: x \in V\} \subseteq W$

# Dual Spaces

## Dual Spaces

• Dual space of $$V$$, $$V^\ast = \{ v^\ast: V \to \mathbb{R} s.t. v^\ast \text{ continuous} \}$$
• Example: $${\mathbb{R}^n}^\ast \simeq \mathbb{R}^n$$
• Prices are elements of a dual space
• Example: $${\mathcal{L}^p}^\ast \simeq \mathcal{L}^q$$ where $$1/p + 1/q = 1$$

## Transpose

• $$A: V \to W$$, the transpose is $$A^T:W^\ast \to V^\ast$$ defined by $(A^T w^\ast)v = w^\ast (Av)$

# Separating hyperplane theorem

## Hyperplanes

• Hyperplane $$H_{\xi,c} = \{x \in V: \xi x = c\}$$ where $$c \in \mathbb{R}$$ and $$\xi \in V^\ast$$

• is affine i.e. $$\forall x, y \in H_{\xi,c}$$ and $$\lambda \in \mathbb{R}$$, $$\lambda x + (1-\lambda)y \in H_{\xi,c}$$

## Convexity

• $$S \subseteq V$$ is convex if $$\forall x, y \in S$$ and $$\lambda \in (0,1)$$, we have $$x \lambda + y(1-\lambda) \in S$$

## Separating hyperplane theorem

• If $$S_1$$ and $$S_2 \subseteq V$$ are convex and $$S_1 \cap S_2 = \emptyset$$ and either $$V$$ is finite dimensional or the interior of $$S_1$$ or $$S_2$$ is not empty. Then there exists a hyperplane, $$H_{\xi c} = \{ x: \xi x = c \}$$ such that $\xi s_1 \leq c \leq \xi s_2$ for all $$s_1 \in S_1$$ and $$s_2 \in S_2$$. We say that $$H_{\xi,c}$$ separates $$S_1$$ and $$S_2$$.

## Lagrange multipliers exist

• Let $$V$$ and $$W$$ be normed vector spaces, $$A \in B(V,\mathbb{R})$$ and $$C \in B(V,W)$$. Assume that $$range(C)$$ is closed. Then $$null(A) \supseteq null(C)$$ if and only if $$A = \mu C$$ for some $$\mu \in W^*$$

## Farkas’ Lemma

Let $$A$$ be an $$m$$ by $$n$$ matrix and $$b \in \mathbb{R}^m$$ either:

1. $$\exists x = (x_1, ..., x_n) \in \mathbb{R}^n$$ such that $$x_1 \geq 0, ..., x_n \geq 0$$ and $$Ax = b$$ or
2. $$\exists y \in \mathbb{R}^m$$ such that $$y^T A e_i \geq 0$$ for each standard basis vector $$e_i \in \mathbb{R}^n$$ and $$y^T b < 0$$

but not both.

# Welfare theorems

## Notation

• Set of commmodities $$S$$, a normed vector space
• $$I$$ consumers with preference relations $$\succeq_i$$
• $$J$$ firms with production possibility sets $$Y_j$$
• Allocation $$(x,y) \in S^{I + J}$$
• Feasible if $$y_j \in Y_j$$ and $$\sum_i x_i = \sum_j y_j$$

## Pareto efficient

• $$(x^0,y^0)$$ Pareto efficient if feasible and no other feasible allocation such that $$x_i \succeq_i x_i^0$$ for all $$i$$ and $$x_i \succ_i x_i^0$$ for at least one $$i$$

## Competetive equilibrium

Allocation $$(x^e,y^e)$$ and price, $$p \in S^*$$ such that

1. $$(x^e,y^e)$$ feasible
2. Consumers maximize $$px \leq p x_i^e$$ implies $$x_i^e \succeq_i x$$
3. Firms maximize, $$y \in Y_j$$ implies $$py \leq p y_j^e$$

## Preference properties

• Local non-satiation : for each $$x$$ and $$\epsilon>0$$ $$\exists x'$$ such that $$\Vert x-x' \Vert \leq \epsilon$$ and $$x' \succ_i x$$
• Convex : if $$x \succeq_i z$$ and $$y \succeq_i z$$, $$\lambda \in [0,1]$$ then $$\lambda x + (1-\lambda) y \succeq_i z$$
• Continuous : for any $$x \succ_i z$$ $$\exists \delta > 0$$ such that for all $$x' \in N_\delta(x)$$, $$x' \succ_i z$$

## First Welfare theorem

If $$(x^e,y^e)$$ and $$p$$ is a competitive equilibrium and preferences a locally non-satiated, then $$(x^e,y^e)$$ is Paretor efficient.

## Second Welfare theorem

If preferences are locally non-satiated, convex, and continuous, and $$(x^e, y^e)$$ is Pareto efficient, then $$\exists p \in S^*$$ such that $$(x^e,y^e)$$ and $$p$$ is a competitive equilibrium.