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{\LARGE List of Theorems and Defined Terms }
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& Function
&& One-to-One
&& onto
& Field
&& Five properties
&& Theorems
&&& Cancellation laws
&&& 0, 1 unique
&&& $a \times 0 = 0$
&&& $(-a) \times b = a \times (-b) = -(a \times b)$, $(-a) \times (-b) = a \times b$
&&& $0^{-1}$ doesn't exist
&&& ...
& Complex Numbers
&& Addition, multiplication, is a field
&& The complex conjugate
&&& The conjugate of the conjugate of z = z
&&& ...
&& Absolute value
&&& $|zw| = |z|\times|w|, |\frac z w| = \frac {|z|} {|w|}, |z+w| \leq |z|+|w|, |z|-|w| \leq |z+w|$
&& $z = e^{i\theta}$
&& Fundamental theorem of algebra (The proof in the book uses things we have not covered):

Suppose that $p(z) = a_nz^n + a_{n-1}z^{n-1} + ... + a_0$ is a polynomial in P(C) of degree $n \geq 1$. Then p(z) has a zero.
& Vector Space
&& 8 Properties
&& Theorems
&&& Cancellation law for vector addition
&&& $0_v$ is unique, -x is unique
&&& $0x = 0, (-a)x = -(ax) = a(-x), a0 = 0$
& Subspaces
&& Four properties
&&& One redundant
&& Theorems
&&& A subset W of V is a subspace of v iff $0 \in W$, $x + y \in W$ if $x \in W$ and $y \in W$, $cx \in W$ if $c \in F$ and $x \in W$
&&& The intersection of subspaces of V is a subspace of V
&&& The union of two subspaces of V is a subspace of V iff one contains the other
& Linear Combinations
&& Span(S)
&&& Theorem: Span(S) is a subspace of V, if a subspace contains S, it contains Span(S)
&&& Generating/Spanning a vector space
& Linear Dependence / Linear Independence
&& Trivial representation of 0
&& linear (in)dependence of $\emptyset, \{0\}, \{a\}: a \neq 0$
&& Theorems
&&& let $S_1 \subseteq S_2 \subseteq V$ If $S_1$ is linearly dependent, then $S_2$ is linearly dependent. If $S_2$ is linearly independent, then $S_1$ is linearly independent.
&&& Let S be a linearly independent subset of V, and let v be a vector in V not in S. Then $S \cup \{v\}$ is linearly dependent iff $v \in span(S)$
& Basis
&& Theorems
&&& let $\beta = \{u_1, u_2,...,u_n\}$ be a subset of V. Then $\beta$ is a basis for V iff all elements in V as a linear combination of vectors in $\beta$ in only one way
&&& If V is generated by a finite set S, then some subset of S is a basis for V. Hence V has a finite basis.
&&& Replacement Theorem: Let V be a vector space that is generated by a set G containing exactly n vectors, and let L be a linearly independent subset of V containing exactly m vectors. Then $m \leq n$ and there exists a subset H of G containing exactly n - m vectors such that $L \cup H$ generates V
&&&& Let V be a vector space having a finite basis. Then every basis for V contains the same number of vectors.
&&&& Let V be a vector space with dimension n:
&&&&& Any finite generating set for V contains at least n vectors, and a generating set for V that contains exactly n vectors is a basis for V.
&&&&& Any linearly independent subset of V that contains exactly n vectors is a basis for V.
&&&&& Every linearly independent subset of V can be extended to a basis for V.

& Dimensions
&& Finite Dimensional, Infinite dimensional
&& Theorems (We haven't gone over these in class yet, I don't know if they are valid subject matter for the term test)
&&& Let W be a subspace of a finite-dimensional vector space V. Then W is finite-dimensional and $dim(W) \leq dim(V)$. Moreover, if $dim(W) = dim(V)$, then $V = W$.
&&&& If W is a subspace of a finite-dimensional vector space V, then any basis for W can be extended to a basis for V.
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