Sec-2 AI Applications Math Review PDF

# Sec-2-AI applications Math_Review ## Vectors, Matrices and Linear Algebra A vector is a member of vector space A Vector is an ordered sequence of values For example: In two-dimensional space, we have vectors such as: * X = (3, 4) and Y = (0, 2) The two fundamental operations on Vectors are:...

# Sec-2-AI applications Math_Review ## Vectors, Matrices and Linear Algebra A vector is a member of vector space A Vector is an ordered sequence of values For example: In two-dimensional space, we have vectors such as: * X = (3, 4) and Y = (0, 2) The two fundamental operations on Vectors are: vector addition and Scalar multiplication. ### Examples * X + Y = (3 + 0, 4 + 2) = (3, 6) * Scalar multiplication, multiplies each element by a constant * 5 X = 5 (3, 4) = (15, 20) The length of a vector is: |X| = √3² + 4² = √25 = 5 ## The dot product also called (Scalar Product) X.Y = Σ x_ⁱ y_ⁱ X.Y = 3x0 + 4x2 = 8 ## A matrix: Is a rectangular array of values arranged into rows and columns ## Example: matrix M of size 3x4 * M_₁₁ M_₁₂ M_₁₃ M_₁₄ * M_₂₁ M_₂₂ M_₂₃ M_₂₄ * M_₃₁ M_₃₂ M_₃₃ M_₃₄ The first index of M_{_ij} specifies the row and the second specifies the column. In programming languages m_{_ij} is often written as m[i, j] or m[i][j] ## The sum of 2 matrices is defined by adding corresponding elements Thus, (m + n)_{_ij} = m_{_ij} + n_{_ij} Note that matrices with the same size. If the matrices are with different size, the sum is undefined. ## Multiplication of matrices, by scalar: (Cm)_{_ij} = Cm_{_ij} ## Examples * 5 * (2 4) = (10 5) * (3 4) = (15 20) * M_{_3,2} = (1 2 3) * N_{_3,2} = (0 1 5 2) * (4 5 6) * (1 2 1) * M + N = (1 7 5) * (5 8 7) * (4 1 3) ## Matrix Multiplication The product mn is defined only if M is size a x b and n is of size b x c. m a x b . n b x c -> is defined mn ≠ nm ## Example: * M_{_3,3} = (1 3 5) * (2 4 6) * (0 1 2) * (2 5 1) * m n = (1 x 0 + 3 x 2 + 5 x 0 + 1 x 1 + 3 x 1 + 5 x 5 + 2 x 2 + 3 x 5 + 5 x 6) * (2 x 0 + 4 x 2 + 6 x 0 + 2 x 1 + 4 x 1 + 6 x 5 + 2 x 2 + 4 x 5 + 6 x 6) * mn = (13 8 35) * (8 35 44) ## The Identity matrix I has elements I_{_i}_{_j} equal to 1 when i = j and equal to zero otherwise. It has the property m I = m ## The transpose of matrix: m_{_ij}^{^T} = m_{_ji} ## Example: * m_{_3x2} = (2 1) * (4 7) * (1 5 8) * m^{^T}_{_2x3} = (2 4 7) * (1 5 8)