Introduction To Linear Algebra(2) Matrix Algebra

properties of invertible matrix

$(AB)^{-1}=B^{-1}A^{-1}$
$(A^T)^{-1}=(A^{-1})^{T}$
$[A\quad I] \rightarrow[I\quad A^{-1}]$
condition number of matrix: $cond(A)=||A||*||A^{-1}||$
big condition number means the matrix is in ill-condition and it maybe not invertible
Here are some statements either all true or all false:
a.A is an invertible matrix
b.A is row equavalent to $n\times n$ identity matrix
c.A has n pivot positions
d.The equation $Ax=0$ has only the trivial solution
e.The columns $A$ from a linear identity set.
f. The linear transform $x\rightarrow Ax$ is one to one
g.The equation $Ax=b$ has at least one solution for each $b$ in $R^n$
h.The columns of $A$ span $R^n$
i.The linear transformation $x\rightarrow Ax$ maps $R^n$ onto $R^n$
j.There is an $n\times n$matrix $C$ such that $CA=I$ or $AC=I$
i. $A^T$ is an invertible matrix

Partitioned Matrices

A matrix of the form $A=\begin{bmatrix}A_{11}&A_{12} \\0&A_{22} \end{bmatrix}$ is said to be block upper triangular. Assume that $A_{11}$ is $p\times p$, $A_{22}$ is $q\times q$ and $A$ is invertible. Then $A^{-1}=\begin{bmatrix}A_{11}^{-1}& -A_{11}^{-1}A_{12}A_{22}^{-1}\\0&A^{-1}_{22}\end{bmatrix}$

Matrix Factorizations

LU Factorization
$A=LU=\begin{bmatrix}1&0&0&0\\*&1&0&0\\*&*&1&0\\*&*&*&1\end{bmatrix} \begin{bmatrix}\times&*&*&*&*\\0&\times&*&*&*\\0&0&0&\times&*\\0&0&0&0&\times\end{bmatrix}$
Then the solution of $Ax=b$ could be get through $Ly=b$ and $Ux=y$

Subspaces of $R^n$

The subspaces of $R^n$ is any set $H$ in $R^n$ that has three propoerties:
a.The zero vector in $H$.
b.For each $u$ and $v$ in $H$, the sum $u+v$ is in $H$
c.For each $u$ in $H$ and each scalar $c$, the vector $cu$ is in H
The Null space of a matrix A is the set Nul A of all solutions of the homogeneous equation $Ax=0$
The Null space of a $m\times n$ matrix A is subspace of $R^n$
Basis: A basis for a subspace H of $R^n$ is a linearly independent set in H that span H
The pivot columns of a matrix $A$ form a basis for the column space of $A$.

Demension and Rank

Demension of a nonzero subspace $H$, denoted by dim $H$, is the number of vectors in any basis for $H$. The demension of the zero subspace {0} is definded to be zero.
The rank of a matrix $A$, denoted by rank $A$, is the demension of the column space of $A$.
If a matrix $A$ has n columns, then $rank A + dim Nul A = n$
Rank and the Invertible matrix theorem
Let $A$ be $n\times n$ matrix. Then the following statements are each equivalent to the statement that $A$ is an invertible matrix:
m. the columns of $A$ form a basis of $R^n$
n. Col A= $R^n$
o. dim Col A=n
p. rank A=n
q. Nul A={0}
r. dim Nul A=0