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Linear equation

A linear equation is an algebraic equation in which each term is a constant or the product of a constant and a single variable to the power of one, with no products of variables or higher powers present. In one variable, it takes the form ax + b = [0](/page/0), where a and b are constants and a \neq 0.^[1] For multiple variables, the general form is a_1 x_1 + a_2 x_2 + \dots + a_n x_n = b, where the a_i are coefficients and b is a constant.^[2] Such equations represent straight lines when graphed in the plane for two variables, and their solutions correspond to points where the variables satisfy the equality.^[3] The study of linear equations forms a foundational part of algebra and has ancient origins, dating back over 4,000 years to Babylonian mathematicians around 2000 BC, who solved simple linear equations using geometric methods.^[4] Ancient Chinese texts, such as The Nine Chapters on the Mathematical Art from around 200 BC, developed systematic approaches to solving systems of linear equations through methods resembling Gaussian elimination.^[5] In the 19th century, Carl Friedrich Gauss advanced the field with his elimination method for systems, which remains a cornerstone of linear algebra.^[6] These developments laid the groundwork for modern applications in diverse fields. Systems of linear equations, consisting of multiple such equations solved simultaneously, are solved using techniques like substitution, elimination, or matrix methods, yielding unique solutions, infinitely many, or none depending on the system's consistency.^[7] In engineering, they model electrical circuits, structural analysis, and control systems by representing balances of forces or currents.^[8] In chemistry, systems of linear equations balance chemical reactions by equating atom counts across reactants and products.^[9] Broader applications span economics for input-output models, computer graphics for transformations, and machine learning for regression analysis, underscoring their versatility in approximating real-world phenomena.^[10]^[11]

Fundamentals

Definition

A linear equation is a polynomial equation of degree one, meaning it can be expressed such that the highest power of any variable appearing in it is 1, with no products of distinct variables or compositions of nonlinear functions applied to variables.^[12] This distinguishes linear equations from higher-degree polynomials, where variables may appear raised to powers greater than one or multiplied together in ways that increase the overall degree. For instance, while a linear equation involves only linear combinations of variables equated to a constant, more complex relations exceed this linearity. In contrast to nonlinear equations, which may involve quadratic or higher terms, linear equations maintain a structure that avoids such complexities; for example, the equation x^2 + y = 1 is nonlinear due to the squared term, leading to a parabolic graph rather than a straight line.^[13] This linearity ensures that the solution set, when it exists, forms an affine subspace within the vector space of variables, which is a flat, translated version of a linear subspace parallel to the solution set of the associated homogeneous equation.^[14]

General Form

A linear equation in one variable x is expressed in the general form ax + b = 0, where a and b are constants with a \neq 0.^[1] Here, a serves as the coefficient of the variable, scaling its contribution, while b represents the constant term that shifts the equation away from the origin. This form embodies the polynomial structure of degree one, ensuring the equation remains linear as per its definition. For two variables x and y, the general form extends to ax + by + c = 0, where a, b, and c are constants, and a and b are not both zero to avoid degeneracy.^[15] In this representation, a and b are the coefficients multiplying the respective variables, determining their relative weights, and c is the constant term. This structure maintains the first-degree polynomial nature across both variables. The form generalizes to n variables x_1, x_2, \dots, x_n as

a_1 x_1 + a_2 x_2 + \dots + a_n x_n + c = 0,

where a_1, \dots, a_n, c are constants and not all a_i = 0.^[2] Each a_i acts as the coefficient for x_i, and c is the constant term, preserving the linear polynomial structure in higher dimensions. To simplify, equations are often normalized by dividing all terms by a leading nonzero coefficient, such as a_1, yielding a monic form where the first coefficient is 1.^[16] This normalization aids in comparisons and computations without altering the solution set.

One Variable

Solution Process

The solution process for a linear equation in one variable, typically expressed in the general form ax + b = 0 where a and b are constants and a \neq 0, involves algebraic manipulations to isolate the variable x.^[17] This method relies on applying the properties of equality—addition, subtraction, multiplication, and division—to both sides of the equation while preserving balance.^[18] To isolate the variable, first simplify the equation by combining like terms if necessary, then use inverse operations to move constants to one side and the variable to the other. For example, consider the equation $2x + 3 = 7. Subtract 3 from both sides to obtain $2x = 4, then divide both sides by 2 to yield x = 2.^[17] This step-by-step approach ensures the variable is solved for systematically.^[18] Special cases arise when the coefficient of the variable is zero. If a = 0 and b \neq 0, such as in $0x + 5 = 0, the equation simplifies to a false statement like $5 = 0, indicating no solution exists.^[19] Conversely, if a = 0 and b = 0, as in $0x + 0 = 0, the equation is true for all values of x, resulting in infinitely many solutions.^[20] In practical scenarios, linear equations often stem from word problems requiring translation into algebraic form. For instance, the statement "twice a number plus five equals eleven" translates to $2x + 5 = 11, where x represents the number; solving yields x = 3./01%3A_Chapters/1.20%3A_Word_Problems_for_Linear_Equations) This translation involves identifying the unknown, expressing relationships with operations, and setting up the equality based on the problem's conditions.^[21] Verification confirms the solution's correctness by substituting the value back into the original equation. For x = 2 in $2x + 3 = 7, substitution gives $2(2) + 3 = 7, or $7 = 7, which holds true; discrepancies indicate errors in solving or setup.^[17]

Applications in One Dimension

Linear equations in one variable are widely used to model scenarios where quantities change at a constant rate, such as in problems involving distance, speed, and time. The fundamental relationship is given by the equation d = r t, where d is distance, r is rate (or speed), and t is time.^[22] For instance, if a vehicle travels 120 miles at a constant speed of 60 miles per hour, the time required can be found by solving the linear equation $60t = 120, yielding t = 2 hours.^[23] This approach simplifies planning for travel or logistics by assuming uniform motion without acceleration or external influences.^[24] In business and economics, one-variable linear equations model cost and revenue functions to determine break-even points, where total revenue equals total costs. A typical cost function might be C(x) = mx + b, with fixed costs b and variable cost per unit m, while revenue is often R(x) = px for price p per unit sold x. Setting R(x) = C(x) leads to a linear equation like $10x = 4000 + 2x, solved as $8x = 4000, so x = 500 units, indicating the production level at which the business neither profits nor loses money.^[25] Such models help entrepreneurs assess viability under constant pricing and production costs.^[26] Linear equations also facilitate conversions between temperature scales, which follow a linear relationship due to fixed reference points and equal interval sizes. The conversion from Celsius (C) to Fahrenheit (F) is expressed as F = \frac{9}{5}C + 32, derived from the scales' definitions where water freezes at 0°C (32°F) and boils at 100°C (212°F).^[27] For example, to convert 25°C to Fahrenheit, solve F = \frac{9}{5}(25) + 32 = 77^\circF. This equation is essential in scientific, engineering, and everyday applications involving international standards.^[28] Despite their utility, linear equations in one dimension have limitations when applied to real-world problems, as they assume constant rates and linearity without accounting for nonlinear factors like acceleration, varying costs, or thresholds. For example, in rate problems, real travel often involves changing speeds due to traffic or terrain, making the constant-rate assumption an approximation rather than exact.^[29] Similarly, cost models may fail if economies of scale introduce nonlinearity. These models provide valuable insights for simple, steady-state scenarios but require more advanced techniques for complex dynamics.^[30]

Two Variables

Algebraic Forms

Linear equations in two variables can be expressed in several algebraic forms, each suited to different contexts such as computation, graphing preparation, or deriving equations from known points. These forms are derived from the general linear equation ax + by + c = 0, where solving for one variable yields specialized representations.^[31] The slope-intercept form is y = mx + b, where m represents the slope of the line and b is the y-intercept, the point where the line crosses the y-axis. This form is obtained by starting with the general equation ax + by + c = 0 and solving for y: divide through by b (assuming b \neq 0) to get y = -\frac{a}{b}x - \frac{c}{b}, identifying m = -\frac{a}{b} and the y-intercept -\frac{c}{b}.^[31]^[32] The point-slope form, y - y_1 = m(x - x_1), is useful when the slope m and a point (x_1, y_1) on the line are known. It arises directly from the definition of slope: the change in y over the change in x from the point (x_1, y_1) to any other point (x, y) on the line equals m, leading to \frac{y - y_1}{x - x_1} = m, which rearranges to the given form.^[33]^[34] The standard form is Ax + By = C, where A, B, and C are constants, often integers with A \geq 0 and \gcd(A, B, C) = 1 to ensure a normalized representation. This form is advantageous for equations involving integer coefficients, facilitating operations like finding intercepts or solving systems without fractions. Conversions between these forms are straightforward algebraic manipulations. For instance, to convert from point-slope form y - y_1 = m(x - x_1) to slope-intercept form, distribute m on the right side to get y - y_1 = mx - mx_1, then add y_1 to both sides: y = mx + (y_1 - mx_1), where the new constant term is the y-intercept b = y_1 - mx_1.^[33]^[35] When two points (x_1, y_1) and (x_2, y_2) are given, the two-point form derives the equation by first computing the slope m = \frac{y_2 - y_1}{x_2 - x_1} (assuming x_2 \neq x_1), then substituting into the point-slope form using one of the points, such as y - y_1 = m(x - x_1). This yields an equation passing through both points without directly specifying the intercept.^[36]^[37]

Geometric Interpretation

In the Cartesian plane, a linear equation in two variables, expressed in the general form ax + by + c = 0 where a, b, and c are constants and not both a and b are zero, represents the set of all points (x, y) that satisfy the equation, which geometrically forms a straight line.^[38] This solution set consists of infinitely many points lying along the line, illustrating how the equation constrains the variables to a one-dimensional subset of the two-dimensional plane.^[39] The slope of this line, denoted m, measures its steepness and direction, and from the general form, it is given by m = -\frac{a}{b} assuming b \neq 0; a positive slope indicates the line rises from left to right, a negative slope falls, zero slope means horizontal, and undefined slope (when b = 0) means vertical.^[40] The x-intercept, where the line crosses the x-axis (y = 0), is the point \left( -\frac{c}{a}, 0 \right) if a \neq 0, and the y-intercept, where it crosses the y-axis (x = 0), is \left( 0, -\frac{c}{b} \right) if b \neq 0; these points provide key coordinates for graphing the line.^[41] Two such lines are parallel if they never intersect and maintain a constant distance apart, which occurs when their slopes are equal (i.e., -\frac{a_1}{b_1} = -\frac{a_2}{b_2} for equations a_1 x + b_1 y + c_1 = 0 and a_2 x + b_2 y + c_2 = 0), and the coefficient triples are not proportional (i.e., there is no scalar k such that a_2 = k a_1, b_2 = k b_1, and c_2 = k c_1).^[42] Conversely, two lines are perpendicular if they intersect at a right angle, corresponding to slopes that are negative reciprocals (i.e., m_1 \cdot m_2 = -1, or \frac{a_1}{b_1} = -\frac{b_2}{a_2} assuming neither is vertical).^[43] An alternative geometric representation is the vector form of the line, which can be expressed as a point on the line plus scalar multiples of a direction vector, such as \mathbf{r}(t) = \mathbf{r_0} + t \mathbf{d}, where \mathbf{r_0} is a position vector to a point on the line and \mathbf{d} is a direction vector parallel to it; this form highlights the line's parametric nature in the plane.^[44]

Linear Functions

A linear function in two variables can be expressed as f(x) = mx + b, where m is the slope and b is the y-intercept, representing a mapping from the domain of real numbers to the range of real numbers. This form defines a function provided it passes the vertical line test, meaning each input x produces exactly one output y, distinguishing it from non-functional relations. The graph of such a linear function appears as a straight line extending infinitely in both directions across the Cartesian plane. If m > 0, the function is increasing, rising from left to right; if m < 0, it is decreasing, falling from left to right; and if m = 0, it is constant, forming a horizontal line. This monotonic behavior ensures that non-constant linear functions are injective, or one-to-one, as distinct inputs always yield distinct outputs when m \neq 0. While piecewise linear functions combine multiple linear segments to approximate more complex behaviors, the focus here remains on single linear functions for their simplicity and foundational role in analysis. In modeling real-world phenomena, linear functions effectively capture constant rates of growth or decay, such as population trends over short periods or steady depreciation of assets, where the output varies proportionally with the input. For instance, in economics, they model cost functions where total cost increases linearly with production quantity.

Multiple Variables

Generalization to n Variables

A linear equation in n variables, where n \geq 1, takes the form \sum_{i=1}^n a_i x_i = b, with coefficients a_1, \dots, a_n \in \mathbb{R} not all zero and b \in \mathbb{R}.^[45] The solution set of this equation consists of all points (x_1, \dots, x_n) \in \mathbb{R}^n that satisfy it, forming an (n-1)-dimensional affine hyperplane in \mathbb{R}^n.^[46] This generalizes the geometric interpretations in lower dimensions, such as the line in two variables serving as a special case of a 1-dimensional hyperplane.^[47] In three variables, the equation a x + b y + c z = d represents a plane in \mathbb{R}^3, which is a 2-dimensional hyperplane.^[48] The vector \mathbf{n} = (a_1, \dots, a_n) is the normal vector to the hyperplane, perpendicular to every vector lying within it; specifically, for any two points on the hyperplane, the direction vector between them is orthogonal to \mathbf{n}.^[48] This normal vector defines the orientation of the hyperplane, ensuring that the equation \mathbf{n} \cdot (\mathbf{x} - \mathbf{x_0}) = 0 holds for a point \mathbf{x_0} on the hyperplane and any \mathbf{x} in it.^[48] The solution set is an affine hyperplane, distinct from a linear hyperplane (or linear subspace), which passes through the origin and corresponds to the case b = 0.^[49] An affine hyperplane is a translate of a linear hyperplane by a fixed vector, preserving parallelism but not necessarily containing the origin.^[50] This distinction is crucial in higher-dimensional geometry, as affine hyperplanes model general flat sections without origin constraints.^[51] Visualizing hyperplanes beyond three dimensions poses significant challenges due to human perceptual limits in perceiving more than three spatial dimensions.^[52] Common approaches involve projecting higher-dimensional hyperplanes onto two- or three-dimensional spaces, which can distort intrinsic properties like flatness or intersections, though techniques such as parallel coordinates or slicing help convey structural insights.^[53] These projections aid conceptual understanding but often require abstract reasoning to interpret the full n-dimensional geometry accurately.^[54]

Systems of Equations

A system of linear equations consists of two or more linear equations involving the same set of variables, where the goal is to find values for the variables that satisfy all equations simultaneously. For instance, in two variables, such a system can be expressed as ax + by = c and dx + ey = f, with constants a, b, c, d, e, f. The solution represents the intersection points of the corresponding hyperplanes defined by each equation in the appropriate dimensional space./11%3A_Systems_of_Equations_and_Inequalities/11.01%3A_Systems_of_Linear_Equations_-_Two_Variables) Graphically, for two variables, each equation represents a line in the plane, and the solution is the point where these lines intersect, provided they are not parallel. If the lines are parallel and distinct, the system has no solution; if they coincide, there are infinitely many solutions along the line. In three variables, the equations depict planes in three-dimensional space, whose common intersection may be a single point, a line, or empty, depending on their relative orientations.^[55]^[56] A system is consistent if it possesses at least one solution, meaning the hyperplanes intersect non-trivially. Unique solutions arise when the equations are linearly independent and the hyperplanes are in general position, not parallel, leading to a single intersection point. For cases with infinitely many solutions, the system is consistent but the equations are linearly dependent, resulting in one or more free variables that parameterize the solution set. These solutions form an affine subspace, expressed in parametric form where free variables are assigned parameters, such as t \in \mathbb{R}, to describe all points satisfying the equations. For example, if a system reduces to a single independent equation in three variables, the solution might be parameterized as x = x_0 + t \mathbf{v}_1 + s \mathbf{v}_2, y and z accordingly, where t and s are parameters and \mathbf{v}_1, \mathbf{v}_2 span the null space directions.^[57]^[58] Consider the following two-equation system in two variables:

\begin{cases} 2x + y = 3 \\ x - y = 0 \end{cases}

Adding the equations yields $3x = 3, so x = 1; substituting into the second equation gives y = 1. Thus, the unique solution is (x, y) = (1, 1), corresponding to the intersection of the non-parallel lines./11%3A_Systems_of_Equations_and_Inequalities/11.01%3A_Systems_of_Linear_Equations_-_Two_Variables)

Properties

Uniqueness and Existence

For a single linear equation in multiple variables, the solution set consists of infinitely many points forming an affine hyperplane in the vector space, provided the equation is consistent.^[19] The equation is inconsistent—and thus has no solutions—only if all coefficients are zero but the constant term is nonzero, such as $0 = 1.^[59] In the modern vector space perspective, this inconsistency arises because the zero functional cannot equal a nonzero constant in the affine space.^[60] For a system of linear equations A\mathbf{x} = \mathbf{b}, where A is the m \times n coefficient matrix, existence of solutions is governed by the Rouché–Capelli theorem: the system has at least one solution if and only if the rank of A equals the rank of the augmented matrix [A \mid \mathbf{b}].^[61] If solutions exist but the rank of A is less than n (the number of variables), there are infinitely many solutions, as the solution set forms an affine subspace of dimension n - \rank(A).^[62] Conversely, if \rank(A) < \rank([A \mid \mathbf{b}]), no solutions exist, corresponding geometrically to a collection of parallel hyperplanes that do not intersect.^[61] Uniqueness occurs precisely when the coefficient matrix A has full column rank, \rank(A) = n, ensuring the solution set is a single point in the vector space.^[63] For square systems where m = n, this is equivalent to the determinant of A being nonzero, as a nonzero determinant implies A is invertible and the unique solution is \mathbf{x} = A^{-1}\mathbf{b}.^[64] In elementary terms, full rank means the equations are linearly independent and span the space without redundancy, avoiding both over- and under-constrained cases.^[60]

Homogeneous Equations

A homogeneous linear equation in two variables is of the form ax + by = 0, where a and b are constants, not both zero.^[65] The solutions to this equation form a line passing through the origin in the plane, as any scalar multiple of a solution (x_0, y_0) also satisfies the equation, yielding all points t(x_0, y_0) for t \in \mathbb{R}.^[66] In the general case, a homogeneous system of linear equations is represented as A \mathbf{x} = \mathbf{0}, where A is an m \times n matrix and \mathbf{0} is the zero vector.^[65] The set of all solutions \mathbf{x} \in \mathbb{R}^n forms the null space (or kernel) of A, which is a subspace of \mathbb{R}^n because it is closed under addition and scalar multiplication, and contains the zero vector as the trivial solution.^[65] The dimension of this null space, known as the nullity of A, is given by n - \rank(A), where \rank(A) is the dimension of the column space of A; this is the rank-nullity theorem.^[67] Non-trivial solutions exist if and only if the nullity is greater than zero, which occurs when \rank(A) < n.^[67] For square matrices (m = n), this is equivalent to \det(A) = 0, indicating that the columns (or rows) of A are linearly dependent.^[68] Linear dependence of the columns means there exists a non-zero vector \mathbf{x} such that A \mathbf{x} = \mathbf{0}, directly linking the existence of non-trivial solutions to the dependence relations among the columns.^[68] In applications, such as finding eigenvalues of a matrix A, the characteristic equation leads to the homogeneous system (A - \lambda I) \mathbf{v} = \mathbf{0}, where non-trivial eigenvectors \mathbf{v} exist precisely when \det(A - \lambda I) = 0, with solutions spanning eigenspaces that are subspaces through the origin.^[69]

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Equations of Lines. Among the standard forms for equations of lines are. • The Two-Intercept Form: ax + by = c. • The Point-Slope Form: y − y0 = m(x − x0).Missing: general | Show results with:general
[45]
Linear Equations with Two Variables - Part 2
The standard form is very useful to find the x and y-intercept. For example, in the case of equation 2x + 4y = 8, when x = 0, the equation becomes 4y = 8.
[46]
[PDF] Parallel and Perpendicular Lines - University of Minnesota
If two lines are parallel, their slopes are equal. • If two lines are perpendicular, their slopes are negative reciprocals. That is, if a line has ...
[47]
Equations of a line - Vectors
So in vector-form each point on the line is given by r(t) = tv+b. This is similar to the slope-intercept form for a line in the plane. You can also think of the ...Missing: linear | Show results with:linear
[48]
[PDF] Systems of Linear Equations - UC Homepages
A hyperplane in Rn is the solution set to a linear equation a1x1 + a2x2 + ··· + anxn = b where at least one coefficient is non-zero. When n = 2 we get ax + by ...
[49]
1.4: Lines, Planes, and Hyperplanes - Mathematics LibreTexts
Sep 2, 2021 · In this section we will add to our basic geometric understanding of Rⁿ by studying lines and planes. If we do this carefully, ...
[50]
[PDF] 11.1 Linear Systems
In general, each linear equation in n variables defines a hyperplane in R n, i.e. a flat of dimension n − 1. For example, a linear equation in six ...
[51]
Plane -- from Wolfram MathWorld
The equation of a plane with nonzero normal vector n=(a,b,c) through the point x_0=(x_0,y_0,z_0) is. n·(x-x_0)=0,. (1). where x=(x,y,z) . Plugging in gives the ...
[52]
hyperplane in nLab
Jul 14, 2022 · By a hyperplane one usually means an affine or linear subspace of an affine space or linear space, respectively, typically required to be a positive dimension ...
[53]
affine geometry - PlanetMath
Mar 22, 2013 · Affine geometry studies the geometric properties of affine subspaces, which are associated with vector spaces, and their incidence structure.
[54]
[PDF] Basics of Affine Geometry - UPenn CIS
It is immediately verified that this action makes L into an affine space. For example, for any two points a = (a1, 1 − a1) and b = (b1, 1 − b1) on L, the ...
[55]
[PDF] Visualization of Surfaces in Four-Dimensional Space - Purdue e-Pubs
Unfortunately it is very hard, if not impossible, for us to visualize objects in high-dimensional space. Therefore, visualization of high-dimensional space ...<|control11|><|separator|>
[56]
The Promise and Challenge of Multidimensional Visualization
With the multidimensional system of Parallel Coordinates multidimensional lines, hyperplanes, smooth hypersurfaces and proximities can be visualized ...
[57]
[PDF] Higher Dimensional Graphics: Conceiving Worlds in Four Spatial ...
Mar 26, 2021 · Over the years, computer graphics and visualization research has attempted to enhance intuitive user experiences of these abstract geometric.
[58]
6.1 Solving Systems of Linear Equations
A consistent system of equations has at least one solution. · An inconsistent system is when the equations represent two parallel lines, where the lines have the ...Missing: interpretation | Show results with:interpretation
[59]
[PDF] 1. Systems of Linear Equations - Emory Mathematics
Infinitely many solutions. This occurs when the system is consistent and there is at least one nonleading variable, so at least one parameter is involved.Missing: parameterization | Show results with:parameterization
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[PDF] Section 1.1 : Systems of Linear Equations
A linear system is consistent if it has at least one . Two matrices are row equivalent if a sequence of transforms one matrix into the other. Note: if the ...
[61]
Systems of Linear Equations
According to this definition, solving a system of equations means writing down all solutions in terms of some number of parameters. We will give a ...
[62]
http://www.batmath.it/corsi_uni/maths_1617/rouche-capelli-english.pdf
[63]
[PDF] SYSTEMS OF LINEAR EQUATIONS - UBC Arts
Sep 5, 2013 · Theorem 1.1 (Existence of row echelon form). Any matrix can be put into row echelon form using Gaussian elimination.
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None
### Summary of Existence and Uniqueness of Solutions to Linear Systems Ax = y
[65]
[PDF] The theorem of Rouché & Capelli - Batmath
A linear systrem of m equations in n unknowns is consistent if and only if the coefficient matrix and the augmented matrix have the same rank, called the rank ...
[66]
[PDF] MATH 246: Chapter 2 Section 3: Matrices and Determinants
In linear algebra matrices are used to solve linear systems of equations. ... determinant will be nonzero if and only if there is a unique solution to the system.
[67]
DET-0060: Determinants and Inverses of Nonsingular Matrices
Note that a unique solution to the system exists if and only if the determinant of the coefficient matrix is not zero. It turns out that a solution to any ...Missing: source | Show results with:source
[68]
[PDF] Subspaces, Basis, Dimension, and Rank - Purdue Math
Definition. Let A be an m×n matrix. The null space of A is the subspace of Rn consisting of solutions of the homogeneous linear system Ax = ~0. It is.
[69]
1.3 Rank and Nullity
, is the set of solutions to the homogeneous linear system A x = 0. You may recall that a standard method to deduce the solutions is to put the matrix A in ...Missing: equations | Show results with:equations
[70]
[PDF] MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix ...
Rank is the dimension of row/column space. Nullity is the dimension of the nullspace. Rank + nullity equals the number of columns in a matrix.
[71]
[PDF] Homogeneous Linear Systems
A homogeneous linear system with coefficient matrix A has non-trivial solutions if and only if the columns of A are linearly dependent. Proof. By theorem ...
[72]
[PDF] On Bases and the Rank-Nullity Theorem - Penn Math
Jul 14, 2015 · If b = 0, the system is called homogeneous. In this case, the solution set is simply the null space of A. Any homogeneous system has the ...Missing: equations | Show results with:equations