Algebra 1 Notes SOL A.6 (4.5) Slope
... Example: Solve 2x + 4 = 2x - 4 The figure to the right shows the graph of each side of the equation. We find that the above equation has no solution because the lines are parallel. We also see that the two lines have the same slope. Conclusion: Two lines with the same slope are parallel. ...
... Example: Solve 2x + 4 = 2x - 4 The figure to the right shows the graph of each side of the equation. We find that the above equation has no solution because the lines are parallel. We also see that the two lines have the same slope. Conclusion: Two lines with the same slope are parallel. ...
engr_123_matlab_lab6
... will reduce a matrix A to its reduced row echelon form null(A,′r′) will find a rational basis for null A rank(A) returns the rank of a matrix A x=A\b solves the linear system Ax = b (A is an m × n matrix; x is an n × 1 column vector; b is an m × 1 column vector) rref([A b]) another way to solve Ax = ...
... will reduce a matrix A to its reduced row echelon form null(A,′r′) will find a rational basis for null A rank(A) returns the rank of a matrix A x=A\b solves the linear system Ax = b (A is an m × n matrix; x is an n × 1 column vector; b is an m × 1 column vector) rref([A b]) another way to solve Ax = ...
Sample Final Exam
... S = p(x) ∈ P3 p(2) − p(1) = 0 Find a basis for this subspace. Answer: Suppose that p(x) = ax2 + bx + c is a polynomial in S. Then, p(2) = 4a + 2b + c and p(1) = a + b + c, so that p(2) − p(1) = 3a + b. Thus, 3a + b = 0, so b = −3a. Thus, we can write p(x) as p(x) = ax2 − 3ax + c = a(x2 − 3x) + c Th ...
... S = p(x) ∈ P3 p(2) − p(1) = 0 Find a basis for this subspace. Answer: Suppose that p(x) = ax2 + bx + c is a polynomial in S. Then, p(2) = 4a + 2b + c and p(1) = a + b + c, so that p(2) − p(1) = 3a + b. Thus, 3a + b = 0, so b = −3a. Thus, we can write p(x) as p(x) = ax2 − 3ax + c = a(x2 − 3x) + c Th ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.