Notes from AMATH 352

## Review for Midterm 1

The following topics (but not limited to) could be on the midterm### A norm

magnitude or the length of a vectorsimply a function (or map) taking vectors in Rm as input and

returning nonnegative real numbers as output.

#### Conditions

Has to satisfy the following conditions#### Max norm

Infinite norm#### 2-norm

Euclidean norm#### 1-norm

Manhattan norm#### p-norm

general case of norm### A linear function

### Linear combinations

### A real linear space

#### Definition

a set of objects V with the following properties#### Another way to show linear space

### The vector space Rn and the space of matrices Rm×n

Rm: Set of all real m-vectorsfor x,y in Rm: x+y, ax are another vector in Rm

(therefore, the set is closed under addition and scalar multiplication)

### The function spaces C 0 , C 1 , Pk

#### C0

#### C1

#### Pk

### A subspace

(in particular, that it must be closed under addition and scalar multiplication, and hence under linear combinations).If S is a subset of a linear space V and S is closed under addition and scalar multiplication, then we say that S is a subspace of V .

### Matrix

Matrix-vector and matrix-matrix multiplication, as well as addition and scalar multiplicationof vectors and matrices

### Linear independence

### A basis for a linear space, the dimension

#### Method

a systematic approach to find a basis is defined in the following steps1. Check that the space S is indeed a real linear space. For example, verify that it is a subspace of a bigger real linear space.

2. Write a general formula characterizing any element of S (an element can be a function, a vector, ...)

3. Identify the independent parameters in the formula.

4. Write a new formula characterizing any element of S as a linear combination of elements where each parameter is multiplying one element. This formula should indicate that these elements span the space S.

5. Check whether these particular elements are linearly independent. If they are linearly independent, these elements form a basis for S. If not, look for a subset of these elements

### The null space N (A) and range or column space R(A) of a matrix

### Matrix and vector transpose, AT

### The Euclidean inner product of vectors u, v ∈ Rm

### The inverse matrix and the identity matrix

### If V is a subspace of Rn then dim(V ) ≤ n

### If A ∈ Rm×n then rank(A) ≤ min(m, n) and dim(N (A)) = n − rank(A)

asdf## MATLAB for Midterm 1

• Defining row vs. column vectors, transpose of vectors or matrices

• Colon notation for subarrays of a matrix, e.g. A(:,j) is the jth column and A(2:4,:) is a 3 × n matrix consisting of rows 2,3,4 of A ∈ Rm×n .

• The difference between * and .* for vector or matrix multiplication.

• How to read a simple Matlab program involving for loops and interpret the results of a program.

• I will not ask you to write a Matlab program, but you should be able to read a simple one.