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QMII Summary – Mathematics


Chapter 1 – Matrices and vectors

1. Matrices
Order of a matrix: its number of rows and columns
aij represents the number we find in the i-th row and the j-th column
Transpose of a matrix: if A is a m x n matrix, then its transpose AT is an n x m matrix
which has the number aij in the j-th row and the i-th column.
Zero matrix: a matrix with all entries equal to zero, indicated by the letter O.
Square matrix: a matrix containing the same number of rows and columns
Diagonal entries: entries a11, a22, a33, …
Identity matrix: a square matrix for which all diagonal entries = 1 and all others equal
0. It is usually indicated by the letter I.


2. Vectors
Vector: matrix with only one column. It is indicated by one lowercase letter.
Dimension: the number of entries (= order for matrices)
Zero vector: a vector with all entries =0
Unit vector: a vector with one entry equal to one and all others equal to zero. The
unit vector of which the i-th entry = 1 is indicated by ei.



3. Operations on vectors
Scalar multiplication: the scalar product of a vector x and a number c is the vector cx
obtained by multiplying each entry of x by c.
Sum of vectors: taking the sum of each pair of corresponding entries (same for a
difference).
Linear combination: 7t + 12r is a linear combination of vectors t and r.


4. Operations on matrices
Sum of matrices: we take the sum of each pair of corresponding entries.
Scalar product of a matrix and a number: we multiply all entries of the matrix by the
scalar.

Rules for matrix addition and multiplication by scalars
A+B=B+A c(A + B) = cA + cB
(A + B) + C = A + (B + C) (c + d)A = cA + dA
A+O=A c(dA) = (cd)A
A + (-A) = 0 1A = A




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, QMII Summary – Mathematics


5. The Product of a Matrix and a Vector
We can multiply a matrix and a vector only if the number of columns of the matrix is
equal to the dimension of the vector.
For an (m x n) matrix A and a x-vector of dimension n, the result will be the vector Ax
of dimension m, of which the i-th entry equals the product of the i-th row of A and
the vector x. This product results from multiplying the entries of the row by the
corresponding entries of the vector and then taking the sum of these products.



6. Properties of the matrix-vector product
A(cx) = cAx
A(x+y) = Ax + Ay
By combining these rules, we can show that A(cx + dy) = Acx + Ady


7. The product of two matrices
The product of a matrix A and a matrix B is the matrix AB for which the entry at
position (I,j) is the product if the i-th row of A and the j-th column of B.
The product of two matrices can only be determined if the number of columns of the
first matrix equals the number of rows of the second one.
 if we multiply a (m x n) matrix by a (n x k) matrix, the result will be a (m x k)
matrix.

Power of a matrix: For a square matrix A we write A2 instead of AA, A3 instead of
AAA,…


8. Properties of the matrix product
A ( B + C ) = AB + AC (AB)C = A(BC)
(A + B)C = AC + BC AI = IA = A
(cA)B = A(cB) = cAB (AB)T = BTAT




Chapter 2 – Systems of Linear Equation

2. Systems of linear equation
A system of linear equations with m equations and n variables is a m x n system.

When the system is rearranged in a such that all terms containing a variable are on
the left-hand side in the same order, and the constant terms are on the right-hand
side, it is said to be in the standard form.




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