Traditional values in Caribbean and African societies often place womens’ value in the context of other men. When women pursue independent careers such as midwives and healers, they could be called “witches.”

Maryse Condé demonstrates this bias in the novel I, Tituba. She writes that “Yao’s love had transformed [Tituba]’s mother”, making her a “young woman.” (Condé 7) In the passage, the womanhood of Tituba’s mother is framed as only being granted when she encounters Yao; in contrast, Mama Yaya’s womanhood exists independently, yet she is viewed as a witch.

Statement

Suppose \(U_1\), \(U_2\), and \(W\) are subspaces of \(V\), such that:

\begin{equation} \begin{cases} V = U_1 \oplus W\\ V = U_2 \oplus W \end{cases} \end{equation}

Prove or give a counterexample that \(U_1=U_2\)

Intuition

The statement is not true. The definition of direct sums makes it such that, \(\forall v \in V\), there exists a unique representation of \(v\) with \(u_{1i}+w_{i} = v\) for \(u_{1j}\in U_1, w_{j} \in W\) as well as another unique representation \(u_{2i} + w_{i}=v\) for \(u_{2j} \in U_{2}, w_{j} \in W\).

Claim

Proof or give a counter example for the statement that:

\begin{align} \dim\qty(U_1+U_2+U_3) = &\dim U_1+\dim U_2+\dim U_3\\ &-\dim(U_1 \cap U_2) - \dim(U_1 \cap U_3) - \dim(U_2 \cap U_3) \\ &+\dim(U_1 \cap U_2 \cap U_3) \end{align}

Counterexample

This statement is false.

Take the following three subspaces of \(\mathbb{F}^{2}\):

\begin{align} U_1 = \qty{\mqty(a \\ 0): a \in \mathbb{F}}\\ U_2 = \qty{\mqty(0 \\ b): b \in \mathbb{F}}\\ U_3 = \qty{\mqty(c \\ c): c \in \mathbb{F}} \end{align}

Statement

Support \(W\) is finite-dimensional, and \(T \in \mathcal{L}(V,W)\). Prove that \(T\) is injective IFF \(\exists S \in \mathcal{L}(W,V)\) such that \(ST = I \in \mathcal{L}(V,V)\).

Proof

Given injectivity

Given an injective \(T \in \mathcal{L}(V,W)\), we desire that \(\exists S \in \mathcal{L}(W,V)\) such that \(ST = I \in \mathcal{L}(V,V)\).

We begin with some statements.

• Recall that, a linear map called injective when \(Tv=Tu \implies v=u\)
• Recall also that the “identity map” on \(V\) is a map \(I \in \mathcal{L}(V,V)\) such that \(Iv = v, \forall v \in V\)

Motivating \(S\)

We show that we can indeed create a function \(S\) by the injectivity of \(T\). Recall a function is a map has the property that \(v=u \implies Fv=Fu\).

Statement

Support \(W\) is finite-dimensional, and \(T \in \mathcal{L}(V,W)\). Prove that \(T\) is injective IFF \(\exists S \in \mathcal{L}(W,V)\) such that \(ST = I \in \mathcal{L}(V,V)\).

Proof

Given injectivity

Given an injective \(T \in \mathcal{L}(V,W)\), we desire that \(\exists S \in \mathcal{L}(W,V)\) such that \(ST = I \in \mathcal{L}(V,V)\).

Creating \(S\)

Define a relation \(S:range\ T\to V\) in the following manner:

\begin{equation} S(v) = a \mid Ta = v \end{equation}