3. Definitions and Operations
Definitions
Vector. Let n be a positive integer and let ℝ denote the set of real numbers, then ℝn is the set of all n-tuples of real numbers. A vector v∈ℝn is an n-tuple of real numbers. The notation "∈S" is read "element of S". For example, consider a vector that has three components:
v=((v_1),(v_2),(v_3))∈(ℝ,ℝ,ℝ)≡ℝ3.
Matrix. A matrix A∈ℝ(m×n) is a rectangular array of real numbers with m rows and n columns. For example, a 3×2 matrix looks like this:
A=[[(a_11),(a_21)],[(a_12),(a_22)],[(a_13),(a_23)]]∈[[ℝ,ℝ],[ℝ,ℝ],[ℝ,ℝ]]≡ℝ(3×2)
Vector Operations
We now define the math operations for vectors. The operations we can perform on vectors u=((u_1),(u_2),(u_3)) and v=((v_1),(v_2),(v_3)) are:
| u+v=((u_1)+(v_1),(u_2)+(v_2),(u_3)+(v_3)) |
| u-v=((u_1)-(v_1),(u_2)-(v_2),(u_3)-(v_3)) |
| α*u=(α*(u_1),α*(u_2),α*(u_3)) |
| u⋅v=(u_1)⋅(v_1)+(u_2)⋅(v_2)+(u_3)⋅(v_3) |
| u×v=((u_2)⋅(v_3)-(u_3)⋅(v_2)) |
| ‖u‖=√(,(u_1)2+(u_2)2+(u_3)2) |
The dot product and the cross product of two vectors can also be described in terms of the angle θ between the two vectors. The formula for the dot product of the vectors is u⋅v=‖u‖⋅‖v‖*cos(θ). We say two vectors u and v are orthogonal if the angle between them is 90○.
The dot product of orthogonal vectors is zero: u⋅v=‖u‖⋅‖v‖*cos(90)=0.
The norm of the cross product is given by ‖u*v‖=‖u‖⋅‖v‖*sin(θ).
The cross product is not commutative: u⨯v≠v⨯u , in fact u×v=-v⨯u
Matrix Operations
We denote by A the matrix as a whole and refer to its entries as (a_ij). The mathematical operations defined for matrices are the following:
| C=A+B⇔(c_ij)=(a_ij)+(b_ij)
Subtraction is the inverse of addition |
| [[(α_1),(α_2),(α_3)],[(β_1),(β_2),(β_3)]]T=[[(α_1),(β_1)],[(α_2),(β_2)],[(α_3),(β_3)]] |
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| The product of matrices A∈ℝ(m⨯n) and B∈ℝ(n⨯l) is another matrix C∈ℝ(m⨯l) given by formula: C=A⋅B⇔(c_ij)=(∑_k=1^n)((a_ij)⋅(b_ij))
[[(a_11),(a_12)],[(a_21),(a_22)],[(a_31),(a_32)]]*[[(b_11),(b_12)],[(b_21),(b_22)]]=[[(a_11)*(b_11)+(a_12)*(b_21),(a_11)*(b_12)+(a_12)*(b_22)],[(a_21)*(b_11)+(a_22)*(b_21),(a_12)*(b_12)+(a_22)*(b_22)],[(a_31)*(b_11)+(a_32)*(b_21),(a_31)*(b_12)+(a_32)*(b_22)]]
Note that the matrix product is not a commutative operation: A*B≠B*A
For Matrix-Vector Product, see here →Matrix-Vector Product
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| By definition, the inverse matrix A(-1) undoes the effects of the matrix A. The cumulative effect of applying A(-1) after A is the identity matrix 1 A(-1)*A=1≡[[1,,0],[,⋯,],[0,,1]]
More about →Computing the Inverse |
| The determinant of a matrix, denoted det(A) or |A|, is a special way to combine the entries of a matrix that serves to check if a matrix is invertible or not. The determinant formulas for 2×2 and 3×3 matrices are |[(a_11),(a_12)],[(a_21),(a_22)]|=(a_11)*(a_22)-(a_12)*(a_21), and [[(a_11),(a_12),(a_13)],[(a_21),(a_22),(a_23)],[(a_31),(a_32),(a_33)]]=(a_11)*[[(a_22),(a_23)],[(a_32),(a_33)]]-(a_12)*[[(a_21),(a_23)],[(a_31),(a_33)]]+(a_13)*[[(a_21),(a_22)],[(a_31),(a_32)]] If the |A|=0 then A is not invertible. If |A|≠0 then A is invertible. |
Additional Topics
Using Matrices for Solving Systems of Equations: Systems of Equations
Suppose we're asked to solve the following system of equations:
{[1*(x_1)+2*(x_2)=5],[3*(x_1)+9*(x_2)=21])
Without a knowledge of linear algebra, we could use substitution, elimination, or subtraction to find the values of the two unknowns (x_1) and (x_2).
Gauss-Jordan elimination is a systematic procedure for solving systems of equations based the following row operations:
(α) Adding a multiple of one row to another row
(β) Swapping two rows
(γ) Multiplying a row by a constant
Basis, Dimensions, Eigenvalues and eigenvectors etc: *extra*
More Cheat Sheets ->