Vector Spaces

• rules of vector space
• The Null Space of a Matrix
• The column Space of a Matrix
• Kernel and Range of a Linear Transformation
• Coordinate Systems
• The Coordinate Mapping
• dimension of $Nul A$ and $Col A$
• row space
• The rank theorem
• Rank and the Invertible Matrix Theorem
• Applications to Linear Difference Equations
• Application to Markove Chains

rules of vector space

for each $u,v$u,v in vector space $V$V. $u+v$u+v, $cu$cu, $cv$cv are in $V$V

The Null Space of a Matrix

the set of $x$x that satisfy $Ax=0$Ax=0 is the null space of the matrix $A$A.

The column Space of a Matrix

The column space of $m \times n$m×n matrix $A$A written as $Col A$ColA is a set of all linear combinations of the columns of $A$A if $A=[a_1 \cdots a_n]$A=[a1​⋯an​], thus
$ColA=span\{a_1 \cdots a_n\}$ColA=span{a1​⋯an​}
Noted:
column space of an $m \times n$m×n matrix $A$A is all of $R^m$Rm if and only if the equation $Ax=b$Ax=b has a solution for each b in $R^m$Rm.

Kernel and Range of a Linear Transformation

For a linear transformation $T$T:$V\rightarrow W$V→W, the kernel of $T$T is a set of vectors $x$x: for each $u$u $T(u)=0$T(u)=0, the range of $T$T is the set of all vectors in $W$W of form $T(x)$T(x) for some $x$x in $V$V.
If this transformation is matrix transformation, then kernel of $A$A is $Nul A$NulA, range of $A$A is $Col A$ColA
$\beta=\{b_1,b_2,\cdots b_n\}$β={b1​,b2​,⋯bn​} is the bais of $H$H if: $\beta$β is a linear independent set and $H=span \{b_1,b_2,\cdots , b_n\}$H=span{b1​,b2​,⋯,bn​}

Coordinate Systems

Suppose $\beta=\{b_1,b_2,\cdots b_n\}$β={b1​,b2​,⋯bn​} is the basis for $V$V and $x$x is in $V$V. The coordinates of $x$x relative to the basis $\beta$β are weights $c_1,\cdots,c_n$c1​,⋯,cn​ such that $x=c_1b_1,\cdots,c_nb_n$x=c1​b1​,⋯,cn​bn​

The Coordinate Mapping

Let $\beta=\{b_1,b_2,\cdots b_n\}$β={b1​,b2​,⋯bn​} be the basis of a vector space $V$V. Then the coordinating mapping $x\rightarrow x_{\beta}$x→xβ​ is a one-to-one linear transformation from $V$V onto $R^n$Rn

dimension of $Nul A$NulA and $Col A$ColA

Pivotal columns of a matrix $A$A form a basis for $ColA$ColA, thus the demension of $Col A$ColA is the number of pivotal columns in $A$A.
The dimension of $Nul A$NulA is the free variables in equation $Ax=0$Ax=0

row space

If two matrices $A$A and $B$B are row equivalent, then their row spaces are the same. If $B$B is in echelon form, the nonzero rows of $B$B form a basis for the row space of $A$A as well as for that of $B$B.

The rank theorem

the rank of $A$A is the dimension of the column space of $A$A.
Dimension of $ColA$ColA and $RowA$RowA for $m\times n$m×n matrix $A$A are equal. $Rank A+dimNulA=n$RankA+dimNulA=n

Rank and the Invertible Matrix Theorem

Let A be an $n \times n$n×n matrix. Then the following statements are each equivalent to the statement that $A$A is an invertible matrix.
m. The columns of $A$A form a basis of $R^n$Rn
n.$Col A=R^n$ColA=Rn
o.$dimColA=n$dimColA=n
p.$rankA=n$rankA=n
q.$NulA=\{0\}$NulA={0}
r.$dimNulA=0$dimNulA=0
in practical, the effective rank of a matrix $A$A is often determined from a singular value decomposition of $A$A.
Let $\beta=\{b_1,b_2,\cdots,b_n\}$β={b1​,b2​,⋯,bn​} and $c=\{c_1,\cdots,c_n\}$c={c1​,⋯,cn​} be the bases of a vector space $V$V. Then there is a unique $n \times n$n×n matrix $\mathop{P}\limits_{C\leftarrow B}$C←BP​ such that $[x]_c=\mathop{P}\limits_{C\leftarrow B}[x]_b$[x]c​=C←BP​[x]b​
The columns of $\mathop{P}\limits_{C\leftarrow B}$C←BP​ are $C-$C− coordinate vecctors of the vectors in the basis $\beta$β. That is,$\mathop{P}\limits_{C\leftarrow B}=[ [b_1]_c[b_2]_c \cdots[b_n]_c]$C←BP​=[[b1​]c​[b2​]c​⋯[bn​]c​]

Applications to Linear Difference Equations

If $a_n \ne 0$an​̸​=0 and if $\{z_k\}${zk​} is given, the equation for all $k$k: $y_{k+n}+a_0y_{k+n-1}+a_1y_{k+n-2}+\cdots+a_ny_k=z_k$yk+n​+a0​yk+n−1​+a1​yk+n−2​+⋯+an​yk​=zk​
has a uniques solution whenever the $y_0,\cdots,y_{n-1}$y0​,⋯,yn−1​ are specified.

The set of $H$H of all solutions of the $n$nth-order homogeneous linear difference equation$y_{k+n}+a_0y_{k+n-1}+a_1y_{k+n-2}+\cdots+a_ny_k=z_k$yk+n​+a0​yk+n−1​+a1​yk+n−2​+⋯+an​yk​=zk​ is an n-demensional vector space

Application to Markove Chains

A vector with nonnegative entries that add up to 1 is called a probability vector. A stochastic matrix is a square matrix whose columns are probability vectors. A markove chain is a sequence of probability vectors $x_0,x_1,x_2,\cdots$x0​,x1​,x2​,⋯, together with a stochastic matrix $P$P, such that:
$x_1=Px_0,x_2=Px_1,\cdots x_n=Px_{n-1}$x1​=Px0​,x2​=Px1​,⋯xn​=Pxn−1​