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

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Algebra - Linear Algebra

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V_{1} + ... + V_{n} is a direct sum if and only if Γ is injective

This is the set {v+u : u ∈ U}. So one translate corresponds to one vector v ∈ V and each of its sums with elements of U.

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.

π(v) = v + U. Basically maps a vector to a translate corresponding to that vector. The quotient map already knows the subspace U, the only input is the vector v.

dim V/U = dim V - dim U

that is a map from a quotient space to a vector space. T(v + nullT) = Tv

(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

based off of a basis of V, one element of dual basis maps one element of basis to 1.

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'(φ)(v) = φ(T(v))

The set of all linear functionals which map every single element of a subspace to 0. Subspace of the dual space, V'.

dimV - dimU

nullT' = (rangeT)^{0} dim null T' = dim null T + dimW - dimV

T is surjective

rangeT' = (nullT)^{0} dim range T' = dim range T

T' is surjective

(M(T))^{t}

Always has an eigenvalue

The minimal polynomial of T is a polynomial multiple of the minimal polynomial restricted to the invariant subspace.

T is not invertible

We can get irreducible quadratic factors whose null space has dimension two. This means that any linear map over an odd dimensional vectors space must have at least 1 real eigenvalue.

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