UAHuntsville Linear Algebara - Nazir Abudiab

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Question

Answer

Matrix that is both upper and lower triangular

Diagonal Matrix

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When one matrix is obtained from the other by elementary column operations they are

Column-equivalent matrices

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Operations performed on the column of a matrix

Elementary Column Operations

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Set of all polynomials of degree ≤ n

P_n

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Set of all continuous functions defined on the real number line.

C(-inf,inf)

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Simplest subspace of a vector space V that is consisting of only the zero vector W = {0}

Zero subspace

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Defined as v = C1u1+C2u2+C3u3+...+Cnun where C1,C2,C3,Cn are scalars.

Linear Combination of Vectors

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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

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Vector space that has a basis consisting of a finite number of vector.

Finite Dimensional Vector

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Number of vectors found in the set which makes up the basis of a vector space

Vector Space Dimension

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(T/F) A vector space consists of four entities: a set of vectors, a set of scalars, and two operations.

True

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(T/F) If W is a subspace of R^2, then W must contain the vector (0,0).

True

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(T/F) Every vector space V contains two proper subspaces that are the zero subspace and itself.

False

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(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

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(T/F) If dim(V) = n, then there exists a set of n-1 vectors in V that will span V.

False

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(T/F) If dim(V) = n, then any set of n+1 vectors must be linearly dependent.

True

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(T/F) If A is an invertible n x n matrix, then Ax=b, has a unique solution for every b.

True

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(T/F) If the determinant of n x n matrix A is nonzero, then Ax=0 has only the trivial solution.

True

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(T/F) If two rows of a square matrix are equal then the determinant of the matrix is not zero.

False

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(T/F) The determinant of a matrix order 1 is the entry of the matrix.

True

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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

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is a set of m equations, each of which is linear in the same n variables.

System of Linear Equations

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is a system of linear equations that has at least one solution.

Consistent System

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is a system of linear equations that have no solution.

Inconsistent System

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is a process of rewriting a system of linear equations in row-echelon form.

Gaussian Elimination

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is a matrix derived from the coefficients and constant terms of a system of linear equations.

Augmented Matrix

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are matrices where one is derived from the other by a finite sequence of elementary row operations.

Row equivalent matrices

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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

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is the Zero matrix 0_mn

Additive inverse matrix

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is a matrix that satisfies the condition A = A^T

Symmetric Matrix

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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

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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

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is an n x n matrix which is obtained from the identity matrix I_n

Elementary Matrix

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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

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is a squared matrix in which all the entries below the main diagonal are zero.

Upper Triangular Matrix

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(T/F) The Identity Matrix is an Elementary Matrix

True

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(T/F) The inverse of an Elementary matrix is an elementary matrix.

True

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(T/F) The Zero Matrix is an elementary matrix.

True

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(T/F) If matrix A can be reduced to an identity matrix, then A is nonsingular matrix

True

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(T/F) If A is a square matrix, then the system of linear equations Ax=b has a unique solution.

False

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(T/F) Every matrix has an Additive Inverse

True

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(T/F) Matrix Multiplication is commutative

False

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(T/F) A 4 x 7 matrix has 4 columns

False

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(T/F) A homogeneous system of four linear equations in four variables is always consistent.

True

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(T/F) The product of a 2 x 3 matrix and a 3 x 5 matrix is a 5 x 2 matrix.

False

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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

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The Equation |lambdaI-A|=0 is called the

Characteristic Equation

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Matrices that are similar diagonal matrices are called

Diagonalizable

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A vector Space V with an inner product is called

Inner product space.

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The Dot Product is also called the

Euclidean Inner Product.

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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

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Given [x]B = P[x]B', the matrix P is called

Transition Matrix from B to B'

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The dimension of the nullspace of a matrix A of size m x n is called

Nullity

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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

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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

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(T/F) The column space of matrix A is equal to the row space of matrix A^T

True

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(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

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(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

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(T/F) Elementary row operations preserve the column space of the matrix A.

False

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(T/F) If u*v< 0 the angle is acute

False

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(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

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(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

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(T/F) If A and B are nxn similar matrices, then they always have the same characteristic polynomial equation.

True

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(T/F) The fat that an nxn matrix A has n distinct eigenvalues does not guarantee that A is diagonalizable.

False

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(T/F) If A is diagonalizable, then it has n linearly independent eigenvectors.

True

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