Hofmann Kunze select theorems and definitions

Chapter 1

Theorem 1 ✅

Equivalent systems of linear equations have the same solutions. Two systems are called equivalent if each equation in each system can be expressed as a linear combination of the equations in the other system.

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Theorem 2 ✅

To each elementary row operation $e$ there corresponds an elementary row operation $e_1$, of the same type as $e$, such that $e(e_1(A))=e_1(e(A))=A$ for each $A$. In other words, the inverse operation (function) of an elementary row operation exists and is an elementary row operation of the same type.

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Definition 1(Definition).

$A$ and $B$ are called row equivalent is $B$ can be obtained from $A$ by a finite sequence of row operations and vice versa.

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Theorem 3 ✅

If $A$ and $B$ are row equivalent, the homogeneous systems $AX=0$ and $BX=0$ have exactly the same solutions.

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Info

Hofmann’s row reduced matrices are just RREFs with the order of rows not being fixed.

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Theorem 4

Every $m\times n$ matrix over the field $\mathbb{F}$ is row equivalent to a row reduced matrix.

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Theorem 5 ✅

Every $m\times n$ matrix $A$ is row equivalent to a row reduced echelon matrix.

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Theorem 6

If $A$ is an $m\times n$ matrix and $m<n$, the homogeneous system of linear equation $AX=0$ has a non-trivial solution.

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Theorem 7

If $A$ is an $n\times n$ matrix, then $A$ is row equivalent to the $n\times n$ identity matrix if and only if the system of equations $AX=0$ has only the trivial solution.

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Theorem 8 ✅

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Theorem 9 ✅

Definition 2(Definition).

An elementary matrix is a matrix obtained by performing a single row operation on the identity matrix.

Let $e$ be an elementary row operation and let $E$ be the $m\times m$ elementary matrix $E=e(I)$. Then, for every $m\times n$ matrix $A$, we have $e(A)=EA$.

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Theorem 10 ✅

Definition 3(Definition).

Let $A$ be an $n\times n$ matrix over the field $\mathbb{F}$. An $n\times n$ matrix $B$ such that $BA=I$ is called the left inverse of $A$; an $n\times n$ matrix $B$ such that $AB=I$ is called the right inverse of $A$. If $AB+BA+I$, $B$ is called the two sided inverse of $A$ and $A$ is said to be invertible.

If $A$ has a left inverse $B$ and a right inverse $C$, then $B=C$.

Proof: Suppose $BA=I$ and $AC=I$. Then,
$B=BI=B(AC)=(BA)C=C$.

If $A$ is invertible, so is $A^{-1}$.
If $A$ and $B$ are invertible, so is $AB$, and $(AB{)}^{-1}=A^{-1}{B}^{-1}$.

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Theorem 11 ✅

An elementary matrix is invertible.

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Theorem 12 ✅

If $A$ is an $n\times n$ matrix, the following are equivalent.

1. $A$ is invertible
2. $A$ is row equivalent to the $n\times n$ identity matrix.
3. $A$ is a product of elementary matrices.

If $A$ is an invertible $n\times n$ matrix and if a sequence of elementary row operations reduces $A$ to the identity, then that same sequence of operations when applied to $I$ yields $A^{-1}$.

Let $A$ and $B$ be $m\times n$ matrices. Then, $B$ is row equivalent to $A$ is and only if $B=PA$ for some invertible $n\times n$ matrix $P$.

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Theorem 13 ✅

For an $n\times n$ matrix $A$, the following are equivalent:

1. $A$ is invertible.
2. $A\mathbf{x}=0$ has only the trivial solution.
3. $A\mathbf{x}=\mathbf{y}$ has a solution $\mathbf{x}$ for every $\mathbf{y}\in {\mathbb{R}}^n$.

A square matrix with either a left or right inverse is invertible.

Let $A=A_1{A}_2..A_k$ where $A_1$ through $A_k$ are $n\times n$ square matrices. Then, $A$ is invertible if and only if each $A_j$ is invertible.

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Chapter 2

Definition of Vector spaces.
Definition of Linear Combinations.
Definition of Subspaces.

Theorem 1

A non empty subset $W$ of $V$ is a subspace of $V$ if and only if for each pair of vectors $\alpha ,\beta$ in $W$ and each scalar $c$ in $F$ the vector $c\alpha +\beta$ is again in $W$.

Theorem 2

Intersection of any collection of subspaces is a subspace.

Theorem 3

The subspace spanned by a non-empty subset 8 of a vector
space V is the set of all linear combinations of vectors in 8.

Theorem 4