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MTH355 ~ Chp 1 Terms

Linear Algebra Terms ~ Chp 1, Test 1

Linear Equation An equation in the variables x and y that can be written in the form of ax + by = c, where a, b, and c are real constants (a and b not both zero). The graph of such an equation is a straight line in the xy plane.
Point/Slope Form y = mx + b, where m is the slope and b is the y-intercept
Linear Equation in n Variables When n = x1, x2, x3, ... , xn, this equation can be written in the form a1x1 + a2x2 + a3x3 + ... + anxn = b, where the coefficients a1, a2, a3, ... , an and b are constants.
Gauss-Jordan Elimination A method involving systematically eliminating variables from equations using elementary transformations.1) Write down augmented matrix.2) Derive REF by creating lead 1, then 0s above and below, col by col, starting with 1st col.3) Write sys of eq.
Matrix A rectangular array of numbers. The numbers in the array are called the elements of the matrix.
Elements The numbers in an array.
Submatrix An array obtained by deleting certain rows and columns of a given matrix.
Square Matrix When the number of rows is equal to the number of columns in a matrix.
Row Matrix A matrix consisting of one row.
Column Matrix A matrix consisting of one column.
Identity Matrix A square matrix with 1s in the diagonal locations (1, 1), (2, 2), (3, 3), etc. and zeros elsewhere. We write In for the n x n identity matrix.
Matrix of Coefficients The coefficients of the variables of an equation placed into a matrix.
Augmented Matrix The coefficients of the variables of an equation together with the constant terms placed into a matrix.
Elementary Transformations Transformations that can be used to change a system of linear equations into another sytem of linear equations that has the same solution by eliminating variables.
Reduced Echelon Form Matrix reduced from an augmented matrix consisting of identity matrix on the L and a column matrix of constant terms on the R.1) 0 rows at bottom2) 1st nonzero is a 13) 1 of each row to R of 1 in previous row4) all elements in column w/ lead 1 r 0
Homogeneous A system of linear equations in which all of the constant terms are zeros.
Trivial Solution In a homogeneous system of linear equations in n variables, a solution of x1 = 0, x2 = 0, x3 = 0, ... , xn = 0.
Vector Space Set of all vectors that are closed under scalar multiplication and vector addition.
Position Vector A line segment directed away from the origin by a certain length and direction.
Initial Point The point a vector moves away from. In OA, this is O.
Terminal Point The point a vector terminates at. In OA, this is A.
n-space (R^n) The set of all sequences similar to (u1, u2, u3, ... , un).
Scalar A number.
Vector Addition Adding the corresponding elements of two vectors.
Scalar Multiplication Multiplying every component of a vector by a scalar.
Closed When vector addition or scalar multiplication is performed, closed means that the resulting element is a member of the space under consideration.
Vector Element of a vector space.
Zero Vector Vector having all zero elements.
Subspace Subset of a vector space (R^n) with all of the algebraic properties of R^n. It is closed under vector addition and scalar multiplication.
Span An arbitrary vector can be written as a linear combination of the vectors that span a space.
Linearly Independent For the set of vectors under consideration, the zero vector can only be obtained by the scalars each being equal to zero.
Basis A set of vectors that spans a space and is linearly independent.
Dimension The number of vectors in a basis is the dimension of the space.
Magnitude of a Vector Length of a vector.
Dot Product A process of assigning a real number to each pair of vectors through multiplying the corresponding elements and then adding those elements together.
Norm Length or magnitude of a vector.
Unit Vector A vector whose norm is one.
Normalizing a Vector The procedure of constructing a unit vector in the same direction as a given vector.
Orthogonal Perpendicular - the angle between two nonzero vectors is a right angle.
Created by: sunrise016