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# Linear Algebra Final

### UAHuntsville Linear Algebara - Nazir Abudiab

Matrix that is both upper and lower triangular Diagonal Matrix
When one matrix is obtained from the other by elementary column operations they are Column-equivalent matrices
Operations performed on the column of a matrix Elementary Column Operations
Set of all polynomials of degree ≤ n P_n
Set of all continuous functions defined on the real number line. C(-inf,inf)
Simplest subspace of a vector space V that is consisting of only the zero vector W = {0} Zero subspace
Defined as v = C1u1+C2u2+C3u3+...+Cnun where C1,C2,C3,Cn are scalars. Linear Combination of Vectors
Subset of a Vector space such as V where every vector in V can be written as a linear combination of the vector found in the subset. Spanning Set
Vector space that has a basis consisting of a finite number of vector. Finite Dimensional Vector
Number of vectors found in the set which makes up the basis of a vector space Vector Space Dimension
(T/F) A vector space consists of four entities: a set of vectors, a set of scalars, and two operations. True
(T/F) If W is a subspace of R^2, then W must contain the vector (0,0). True
(T/F) Every vector space V contains two proper subspaces that are the zero subspace and itself. False
(T/F) If a subset S spans a vector space V, then every vector in V can be written as a linear combination of the vector in S. True
(T/F) If dim(V) = n, then there exists a set of n-1 vectors in V that will span V. False
(T/F) If dim(V) = n, then any set of n+1 vectors must be linearly dependent. True
(T/F) If A is an invertible n x n matrix, then Ax=b, has a unique solution for every b. True
(T/F) If the determinant of n x n matrix A is nonzero, then Ax=0 has only the trivial solution. True
(T/F) If two rows of a square matrix are equal then the determinant of the matrix is not zero. False
(T/F) The determinant of a matrix order 1 is the entry of the matrix. True
has the form a1x1+a2x2+a3x3+...+anxn = b, where the coefficients a1,a2,a3,...,an are real numbers, and the constant term b is a real number. Linear Equation in n variables
is a set of m equations, each of which is linear in the same n variables. System of Linear Equations
is a system of linear equations that has at least one solution. Consistent System
is a system of linear equations that have no solution. Inconsistent System
is a process of rewriting a system of linear equations in row-echelon form. Gaussian Elimination
is a matrix derived from the coefficients and constant terms of a system of linear equations. Augmented Matrix
are matrices where one is derived from the other by a finite sequence of elementary row operations. Row equivalent matrices
is a rectangular array in which each entry, a_ij, is a number. Its size is determined by the number of its columns and rows. Matrix
is the Zero matrix 0_mn Additive inverse matrix
is a matrix that satisfies the condition A = A^T Symmetric Matrix
is an n x n matrix when there exists matrix B whose size is n x n such that AB = BA = I_n. Nonsingular Matrix
is the process of writing the n x 2n matrix that consists of a given matrix A on the left and the n x n identity matrix I on the right to obtain [A,I]. Adjoining Matrix I to A
is an n x n matrix which is obtained from the identity matrix I_n Elementary Matrix
is the process of writing a squared matrix A as the product of a lower triangular matrix L and an upper triangular matrix U. LU Factorization
is a squared matrix in which all the entries below the main diagonal are zero. Upper Triangular Matrix
(T/F) The Identity Matrix is an Elementary Matrix True
(T/F) The inverse of an Elementary matrix is an elementary matrix. True
(T/F) The Zero Matrix is an elementary matrix. True
(T/F) If matrix A can be reduced to an identity matrix, then A is nonsingular matrix True
(T/F) If A is a square matrix, then the system of linear equations Ax=b has a unique solution. False
(T/F) Every matrix has an Additive Inverse True
(T/F) Matrix Multiplication is commutative False
(T/F) A 4 x 7 matrix has 4 columns False
(T/F) A homogeneous system of four linear equations in four variables is always consistent. True
(T/F) The product of a 2 x 3 matrix and a 3 x 5 matrix is a 5 x 2 matrix. False
If A is an nxn matrix with eigenvalue lambda, then the set of all eigenvectors of lambda, together with the zero vector is a subspace R^n. The subspace is called ______ of lambda. Eigenspace
The Equation |lambdaI-A|=0 is called the Characteristic Equation
Matrices that are similar diagonal matrices are called Diagonalizable
A vector Space V with an inner product is called Inner product space.
The Dot Product is also called the Euclidean Inner Product.
A procedure in which you are given the coordinates of a vector relative to basis B and you are asked to find another basis B' Change of Basis
Given [x]B = P[x]B', the matrix P is called Transition Matrix from B to B'
The dimension of the nullspace of a matrix A of size m x n is called Nullity
Let B = {c1,v2,v3,...,vn} be an ordered basis for a vector space V and let x be a vector in V such that x = c1v1+c2v2+c3v3+...+cnvn, the scalars c,c2,c3,cb are called Coordinates of x relative to basis
If A is an m x n matrix, then the set of all solutions of the homogeneous system of linear equations Ax=0 is a subspace of R^n called Nullspace
(T/F) The column space of matrix A is equal to the row space of matrix A^T True
(T/F) If a is an mxn matrix of rank r then the dimension of the solution space of Ax=0 is m-r False
(T/F) To perform the change of basis from nonstandard basis B' to the standard basis B the transition matrix P^-1 is simply B' True
(T/F) Elementary row operations preserve the column space of the matrix A. False
(T/F) If u*v< 0 the angle is acute False
(T/F) Geometrically if lambda is an eigenvalue of a matrix A and x is an eigenvector of A corresponding to lambda, then multiplying x by A produces a vector labmda(x) parallel to x. True
(T/F) If A is an nxn matrix with eigenvalue lambda then the set of all eigenvectors lambda is a subspace of R^n False
(T/F) If A and B are nxn similar matrices, then they always have the same characteristic polynomial equation. True
(T/F) The fat that an nxn matrix A has n distinct eigenvalues does not guarantee that A is diagonalizable. False
(T/F) If A is diagonalizable, then it has n linearly independent eigenvectors. True
Created by: evinsmc