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  2. Linear equation - Wikipedia

    en.wikipedia.org/wiki/Linear_equation

    Conversely, every line is the set of all solutions of a linear equation. The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding ...

  3. System of linear equations - Wikipedia

    en.wikipedia.org/wiki/System_of_linear_equations

    When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence. The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent. For example ...

  4. Consistent and inconsistent equations - Wikipedia

    en.wikipedia.org/wiki/Consistent_and...

    The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...

  5. Operator norm - Wikipedia

    en.wikipedia.org/wiki/Operator_norm

    If = {} then the sets in the last two rows will be empty, and consequently their supremums over the set [,] will equal instead of the correct value of If the supremum is taken over the set [ 0 , ∞ ] {\displaystyle [0,\infty ]} instead, then the supremum of the empty set is 0 {\displaystyle 0} and the formulas hold for any V . {\displaystyle V.}

  6. Slope - Wikipedia

    en.wikipedia.org/wiki/Slope

    Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.

  7. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    Equation is called the characteristic equation or the secular equation of A. The fundamental theorem of algebra implies that the characteristic polynomial of an n -by- n matrix A , being a polynomial of degree n , can be factored into the product of n linear terms,

  8. Equality (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Equality_(mathematics)

    Each value of the unknown for which the equation holds is called a solution of the given equation; also stated as satisfying the equation. For example, the equation x 26 x + 5 = 0 {\displaystyle x^{2}-6x+5=0} has the values x = 1 {\displaystyle x=1} and x = 5 {\displaystyle x=5} as its only solutions.

  9. Linear map - Wikipedia

    en.wikipedia.org/wiki/Linear_map

    In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.