Matrix

Question 1

matrix

Answer 1

• a table of numbers
• by convention, dimensions are defined in row-major order
• in other words, as a “row” by “column” matrix
  
  • ex: 2 x 3 matrix
• starts from top-left like in programming
• can be used to represent grid-like data
• can also be used to solve algebraic expressions

Question 2

augmented matrix

Answer 2

• using a matrix to represent a linear system of equations
• each row represents an equation
• each column represents coefficients of a variable or a constant (x, y, etc.)
• don’t need to explicity write x, y or =

Question 3

matrix row operations

Answer 3

• Operations that you can use when solving equations
• Produces an equivalent system of equations
• Essentially the same operations that are allowed in normal algebra

3 Operations

1. Switch any two rows:
   
   • R​₁​ ​↔ R​₂ : Switch row 1 and row 2
2. Multiply a row by a non-zero constant (a non-zero scaler)
   
   • 3R​₂​ ​→ R​₂​​ : Multiply row 2 by 3
3. Add 1 row to another
   
   • R​₁ ​​+ R​₂​ ​→ R​₂ : Add row 1 and 2 into row 2

Question 4

matrix addition (subtraction)

Answer 4

• Add and subtract elements in corresponding cells
• Matrices must have same dimensions; otherwise A+B is not defined
• Order doesn’t matter

Question 5

multiplying by a scalar

Answer 5

• Simply multiply every cell by some constant

Question 6

vector

Answer 6

• (for my purposes) simply a sequence of numbers
• or a matrix with one row or one column (or both, if it’s one cell)
• or an ordered set of numbers
• in this context, equvalent to a tuple, an n-tuple (a tuple with n elements) or a list

Question 7

dot product

Answer 7

• way of combining two vectors of equal length and producing one number
• defined both algebraicly and geometrically
• both are functionally equivalent

Steps (algebraic dot product)

1. Take two vectors of equal length
2. Multiply each term in the same position together
3. Add everything up

Question 8

matrix multiplication

Answer 8

• Convention for multiplying two matrices
• Given matrices A, B
• Each cell (i, j) in final matrix is dot product of A’s row i and B’s column j
• If A, B have have respective dimentions (r_A x c_A) and (r_B x c_B)
• Can only multiply if carb, c_A == r_B, otherwise A*B is undefined
• Final dimensions of new matrix are rax-cub (r_A x c_B)
• Matrix multiplication is not commutative, the order of multiplication matters
• A*B != B*A