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zX9`;Z9Ub{V@T9h&8ve}qA8|aO%5y7-i2Z@H$;ZUN!Sg7PU&DUZ^Op|)N6kTF2=w&V zFy#At7=y~^C3OG|(tn#te|J#*sU(C!3c||P^ifAVwcX#&v8v|AuEq`^oooUE#RLiD z??tgPSU@o1%+{fkeL4w#>LF`kGe54a>b4u`{#&Ge0JF*8eD%nUVFMvS$XR1VK3eUIu9PW&ej$ zH2IA`W-hM(=xd-RdNfP^M#lnDy#K5l3nL3VXpQkZ9TV&0*!eTe$pHvLSOx**K7ouY znR^&Nju6mDRkXDQEz&@e{%IS1!s32fgggy8NfBZ#>i A_1 -> dots -> A_(k-1) -> B$ +- Exists $E_1, dots, E_k$ s.t. $E_k dots E_1 A = B$ +- $M = E_k dots E_1$, $M$ is invertible, $M A = B, A = M^(-1) B$ + +#theorem[Row Equivalence Preserves the linear relations on the columns +- Suppose $A = mat(a_1, dots, a_n)$ and $B = mat(b_1, dots, b_n)$ +- $a_j = c_1 a_1 + dots + c_n a_n arrow.l.r.double b_j = c_1 b_1 + dots + c_n b_n$ +] +#remark[Let $R$ be REF of $A$ +- Pivot columns of $R$ form basis for column space of $R$ +- Columns of $A$ which correspond to pivot columns of $R$ form a basis for column space of $A$ +- No of pivot columns of $R$ is the dim of column space of $A$ +] + +#example[Find basis for a vector space V + +Let $V = "span"{v_1, v_2, v_3, v_4, v_5, v_6}, - v_1 = (1,2,2,1), v_2 = (3,6,6,3), "etc"$ +1. Method 1 (Row Form) + - View each $v_i$ as a row vector + - $mat(1, 2, 2, 1; 3, 6, 6, 3; 4,9,9,5;-2,-1,-1,1;5,8,9,4;4,2,7,3) attach(arrow.r.long, t: "Gaussian Elimination") mat(1,2,2,1;0,1,1,1;0,0,1,1;0,0,0,0;0,0,0,0;0,0,0,0)$ + - $V$ has basis${(1,2,2,1), (0,1,1,1), (0,0,1,1)}$ + - $dim(V) = 3$ +2. Method 2 (Column Form) + - View each $v_i$ as a column vector + - $mat(1,3,4,-2,5,4;2,6,9,-1,8,2;2,6,9,-1,9,7;1,3,5,1,4,3) attach(arrow.r.long, t: "Gaussian Elimination") mat(1,3,4,-2,5,4;0,0,1,3,-2,-6;0,0,0,0,1,5;0,0,0,0,0,0)$ + - Using Pivot columns (1,3,5), pick the corresponding vectors, $(v_1, v_3, v_5)$ + - V has basis${(1,2,2,1), (4,9,9,5), (4,2,7,3)}$ + - $dim(V) = 3$ +] + +#remark[Finding Basis TLDR ++ Method 1: View each $v_1, dots, v_k$ as a row vector + - Find row echelon form $R$ of $vec(v_1,dots.v,v_k)$ + - Nonzero rows of $R$ form basis of $V$ ++ Method 2: View each $v_1, dots, v_k$ as a column vector + - Find row echelon form $R$ of $mat(v_1,dots,v_k)$ + - Find pivot columns of $R$ + - Corresponding columns $v_j$ form basis for $V$ +] + +#example[Extend $S$ to a basis for $RR^n$ +- Find REF of $S$ as row vectors +- Insert the $e_i$ vectors for each $i$ non-pivot column +] + +#theorem[Consistency + +Let $A$ be a $m times n$ matrix +- Column space of $A$ is ${A v | v in RR^n}$ +- Linear system $A x = b$ is consistent + - #sym.arrow.l.r.double $b$ lies in the column space of $A$ +] + +#theorem[Rank + +- Let $A$ be a matrix, then $dim("row space of " A) = dim("col space of " A)$ +- This is the rank of $A$, denoted by $"rank(A)"$ +] + +#remark[ +Let $A$ be an $m times n$ matrix +- $"rank"(A) = "rank"(A^T)$ +- $"rank"(A) = 0 arrow.l.r.double A = 0$ + - $"rank"(A) <= min{m,n}$ + - Full rank if $"rank"(A) = min{m,n}$ +- Square Matrix $A$ is full rank #sym.arrow.l.r.double $A$ is invertible. +] + +#remark[Rank and Consistency of Linear System + +$A x = b$ be linear system +- $A x = b$ is consistent + - $arrow.l.r.double b in "span"{c_1,dots,c_n}$ + - $arrow.l.r.double "span"{c_1,dots,c_n} = "span"{c_1,dots,c_n,b$ + - $arrow.l.r.double dim("span"{c_1,dots,c_n}) = dim("span"{c_1,dots,c_n,b})$ + - $arrow.l.r.double "rank"(A) = "rank"(A|b)$ +- $"rank"(A) <= "rank"(A | b) <= "rank"(A) + 1$ +] + +If $A B x = b$ has a solution, then $A x = b$ also has a solution + + +#theorem[Let $A = m times n$ matrix and $B = n times p$ + +- Column space of $A B subset.eq $ column space of $A$ +- Row space of $A B subset.eq $ row space of $B$ +- $"rank"(A B) <= min{"rank"(A), "rank"(B)}$ +] + +#definition[Nullspace and Nullity + +- Nullspace of $A$ is the solution space of $A x = 0$ + - $v in RR^n | A v = 0$ +- Vectors in nullspace are viewed as column vectors + +- Let $R$ be a REF of $A$ + - $A x = 0 arrow.l.r.double R x = 0$ + - $"nullity"(A) = "nullity"(R)$ +] + +#theorem[Dimension Theorem +Let $A$ be a $m times n$ matrix +- $"rank"(A) + "nullity"(B) = n$ ($n$ is the number of columns) +] diff --git a/ma1522/ch_05.pdf b/ma1522/ch_05.pdf new file mode 100644 index 0000000000000000000000000000000000000000..715456f84ca783c000641a16837acbf2da7f5d54 GIT binary patch literal 170640 zcmeFZby!@@vM-#35P}mt=mdfXnSmMHEf6HQJA=CfhX8?uV8PwpA-KCc!5tFZ-QgR^ zyU%|2-tRv5obS2!{B!R--HTOS)xD~!t5=@Pucj#83yIJJ8Ca1hKEggoNGt#*fTgYp z5*HT|0Kh2kWCaBSQlzz 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b/ma1522/ch_05.typ @@ -0,0 +1,12 @@ +#import "@preview/ctheorems:1.1.2": * +#show: thmrules +#let theorem = thmbox("theorem", "Theorem", fill: rgb("#eeffee"), padding: (top: 0em, bottom: 0em)).with(numbering: none) +#let remark = thmbox("remark", "Remark", padding: (top: 0em, bottom: 0em)).with(numbering: none) +#let definition = thmbox("definition", "Definition", inset: (x: 1.2em, top: 1em), padding: (top: 0em, bottom: 0em)).with(numbering: none) +#let example = thmplain("example", "Example", padding: (top: 0em, bottom: 0em)).with(numbering: none) +#let proof = thmproof("proof", "Proof", padding: (top: 0em, bottom: 0em)) + +#definition[Dot Product. Let $u, v = (u_1, u_2), (v_1, v_2) in RR^2$ + +$u dot v = u_1 v_1 + u_2 v_2$ +]