Linear Algebra Weeks 1-3

What is a field?

-a set K together with two functions (multiplication and addition)

-satisfies nine axioms

What are the four axioms for addition that define a field?

-$\forall a,b,c\in K,a+\left(b+c\right)=\left(a+b\right)+c$ (associativity)

-$\forall a,b\in K,a+b=b+a$ (commutativity)

-$\forall a\in K,a+0=a=0+a$

-$\forall a\in K,\exists b\in K,a+b=0=b+a$

What are the four axioms for multiplication that define a field?

-$\forall a,b,c\in K,a\left(bc\right)=\left(ab\right)c$ (associativity)

-$\forall a,b\in K,ab=ba$ (commutativity)

-$\forall a\in K,1a=a=a1$

-$\forall a\in K\vert\{0\rbrace,\exists b\in K,ab=1=ba$ (no zero in K)

What is the final axioms that define a field?

$\forall a,b,c\in K,a\left(b+c\right)=ab+ac$ (distributivity)

What are examples of fields?

$$\mathbb{C},\mathbb{R},\mathbb{Q}$$

For $a,b\in\mathbb{Z}$, what does a is congruent to b modulo mean? ($a\equiv b\bmod n$ )

$$n\vert\left(a-b\right)\left(i.e.\exists t\in\mathbb{Z},s.t.\left(a-b\right)=tn\right)$$

What is the Division Theorem?

If $n\in\mathbb{N}\land a\in\mathbb{Z},\exists$ a unique $r\in\left\lbrace0,1,2,\ldots,n-1\right\rbrace,a\equiv r\bmod n$

Why do we say that $\mathbb{Z_{p}}$ is a field of characteristic p?

1+1+1+…+1 (p times) = 0

If A is a matrix, when do we say A is symmetric?

if A^T=A

If A is a matrix, when do we say A is skew-symmetric?

if A^T=-A

What does it mean if matrices A and B are row equivalent?

if matrix B can be obtained from matrix A by performing a sequence of elementary row operations

What is row echelon form (REF)?

• any row of zeroes is at the bottom

• in each non-zero row, the pivot is strictly to the left of the pivot in each of the rows below

What is reduced row echelon form (RREF)?

• it is in REF

• each pivot is 1

• all entries above and below a pivot are 0

If $A,B\in M_{n,m}\left(K\right)$, when are A and B row equivalent?

iff RREF(A) = RREF(B)

If $A\in M_{n}\left(K\right)$ what 5 statements are equivalent?

• A is invertible

• The only solution to the matrix equation Ax_ = 0 is x_ = 0

• RREF(A) = $I_{n}$

• A is row equivalent to $I_{n}$

• A can be written as a product of elementary matrices

If A is invertible, what can we say about the EROs which transform A into $I_{n}$?

they will transform $\left(A\vert I_{n}\right)$ into $\left(I_{n}\vert A^{-1}\right)$

When is a matrix L said to be lower triangular?

if L_{i,j}=0,\forall i<j

When is a matrix $\mu$ said to be upper triangular?

if \mu_{i,j}=0,\forall i>j

When is A called an LU decomposition of A?

if A = LU where L is lower triangular and U is upper triangular

When do we say A = LU? (A = $\left(E_{\rho1}^{-1}\ldots E_{\rho t}^{-1}\right)$U)

if we can reduce A to a REF, U, using only row operations of the form:

• $r_{i}\rightarrow r_{i}+\lambda r_{j}$ for j<i,\lambda\ne0