If two lines are intersected by a transversal and corresponding angles are equal in measure → the lines are parallel Converse of Alternate Interior Angles Theorem Congruent complements theorem Same-side Interior Angles Theorem Converse of Corresponding Angles Postulate

Converse of Corresponding Angles Postulate

If two parallel lines are intersected by a transversal, then alternate interior angles are equal in measure Converse of Corresponding Angles Postulate Alternate Exterior Angles Theorem Linear Pair Theorem Alternate Interior Angles Theorem

Alternate Interior Angles Theorem

V_{1} + ... + V_{n} is a direct sum if and only if Γ is injective M(T') = Invariant subspaces and the minimal polynomial Constant term of the minimal polynomial is 0 Let V_{1}, ..., V_{m} be a subspace of V. Define Γ ∶ V_{1} x ... x V_{n} --> V_{1} + ... + V_{n} by Γ (v_{1}, ..., v_{m}) = v_{1} + ... + v_{m}

Let V_{1}, ..., V_{m} be a subspace of V. Define Γ ∶ V_{1} x ... x V_{n} --> V_{1} + ... + V_{n} by Γ (v_{1}, ..., v_{m}) = v_{1} + ... + v_{m}

v-w∈U. Basically the only parts of v and w that are outside of U are in each other, so their translates contain exactly the same vectors. The range of T' Dual basis of V' Linear map over finite complex nozero vector space Equality of translates of a subspace

Equality of translates of a subspace

(a) T squiggly composed with the quotient map = T. (b) T squiggly is injective (c) range of T squiggly = rangeT (d) V/(nullT) and rangeT are isomorphic vector spaces (Their dimensions are the same) DimV' = if we are on the real vector space Null space and range of T squiggly Invariant subspaces and the minimal polynomial

Null space and range of T squiggly

The dual basis gives the coefficients for linear combination: v = φ(v)v_{1} + ... + φ(v)v_{n}. Basically we can construct any vector in v from a linear combination of bases with the dual basis as the scalars. T' is injective The null space of T' Dual basis and linear combinations The range of T'

Dual basis and linear combinations

slope intercept form Cannot be determined All of the above y=mx+b, where m is the slope and b is the y-intercept of the line (0,b) None of the above

y=mx+b, where m is the slope and b is the y-intercept of the line (0,b)

How was Euclid important to geometry? All of the above None of the above Cannot be determined He set up a deductive system by starting with a set of statements (axions), assumed to be true, and then showed that geometric discoveries followed logically from these assumptions.

He set up a deductive system by starting with a set of statements (axions), assumed to be true, and then showed that geometric discoveries followed logically from these assumptions.

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