9/15: MATLAB Basics - Vectors, Colon Operator, Indexing, and Dot Product

Data series and length

• A sequence is a data series; understanding its length is essential for operations on it.

Vector creation and sequences

• Colon operator: $t = start:step:end$ creates a vector from start to end with uniform increments.
  • Example: $t = 0:0.5:2$ → $[0,\, 0.5,\, 1,\, 1.5,\, 2]$ (five points)
• Alternative: $ext{linspace}(a,b,N)$ creates N evenly spaced points from $a$ to $b$ (inclusive).
  • Example: $ext{linspace}(2,4,5) = [2,\, 2.5,\, 3,\, 3.5,\, 4]$
• When using colon operator, the meaning of the last value and increment matters:
  • If you specify a fixed end (e.g., 2 or 4) with a given step, MATLAB stops at or before the end value.
  • If you change the end value, the number of elements may change accordingly.

End-point behavior and counting

• Using a start:step:end with odd/even bounds changes how many elements you get:
  • Example: odd numbers 1:2:20 yields $[1,3,5,7,9,11,13,15,17,19]$ (last is 19, since 20 would exceed the step from 1).
  • If you use 1:2:21, you get one more element (includes 21).
• This is why the last number can affect the count but not the pattern (odd numbers in this case).

Indexing: extracting columns and rows

• For a matrix $A$, you can extract:
  • Entire column j: $A(:,j)$
  • Entire row i: $A(i,:)$
• Example usage patterns:
  • To get all elements of column 3: $A(:,3)$
  • To get all elements of row 1: $A(1,:)$

Basic matrix operations and dot product

• Multiplying a row by a column gives a scalar (dot product):
  • If $A$ is $1 imes N$ and $B$ is $N imes 1$, then $A imes B'$ is a scalar.
• Example (dot product):
  • Let $A = [1 \, 2 \, 0]$ and $B = [0 \, 2 \, 1]^ op$,
    then $A imes B' = 1<em>0 + 2</em>2 + 0*1 = 4.$

Zeros and ones (common utilities)

• Create matrices of zeros or ones for placeholders or masking:
  • $Z = ext{zeros}(m,n)$
  • $O = ext{ones}(m,n)$

Quick takeaways

• Use the colon operator for simple sequences and linspace for exact count control.
• Indexing with (:, j) and (i, :) is the core pattern for column/row extraction.
• Dot products are achieved via multiplying a row vector by a column vector (or using the transpose).