www.YoYoBrain.com - Accelerators for Memory and Learning Questions for Linear Algebra Category: Default - (148 questions) Linear Algebra: Wronskian of a set of Let (y1, y2, ... yn} be a set of functions which function have n-1 derivatives on an interval I. The determinant:| y1, y2, .... yn || y1', y2', ... yn'||.......................|| y1 (n-1 derivative)..... yn (n-1 derivative) | Linear Algebra: Wronskian test for linear Let { y1, y2, .... yn } be a set of nth order independence linear homogeneous differential equations.This set is linearly independent if and only if the Wronskian is not identically equal to zero Linear Algebra: define the dimension of a if w is a basis of vector space then the vector space number of elements in w is dimension Linear Algebra: if A is m x n matrix define row space - subspace of R^n spanned by row and column space of matrix row vectors of Acolum space subspace of R^m spanned by the column vectors of A Linear Algebra: if A is a m x n matrix, then have the same dimension the row space and the column space of A _____ Linear Algebra: if A is a m x n matrix, then have the same dimension the row space and the column space of A _____ Linear Algebra: define the rank of matrix dimension of the row ( or column ) space of a matrix A is called the rank of Arank( A ) Linear Algebra: how to determine the row put matrix in row echelon form and the space of a matrix non-zero rows form the basis of row space of A Linear Algebra: if A is an m x n matrix of rank n x r r, then the dimension of the solution space of Ax=0 is _____ Linear Algebra: define the coordinate Let B = {v1, v2, ... vn } be a basis for vector representation relative to a basis space V and x a vector in V such that:x = c1 v1 + c2 v2 + .... cn vnThen the scalars c1, c2, ... , cn are called coordinates of x relative to B Linear Algebra: a set of vectors s = { v1, v2, the following conditions are true1.) S spans .... , vn } in a vector space V is called a basis V2.) S is linearly independent for v if _____ Linear Algebra: the length of a vector in R^nv || v || = square root ( v1^2 + v2^2 + ... + vn^2 = (v1, v2, ..., vn) ) Linear Algebra: unit vector in the direction of V / || V ||length of 1 and direction of V V Linear Algebra: normalizing the vector V process of finding the unit vector in the direction VV / || V || Linear Algebra: a linear equation in n a1 x1 + a2 x2 + ... + an xn = bThe variables x1, x2, ... , xn has the form coefficients a1, a2, ..., an are real numbers Linear Algebra: a system of linear equations inconsistent is called ___ if it has no solutions Linear Algebra: agumented matrix matrix derived from coefficients and constant terms of a system or linear equations Linear Algebra: 2 properties of matrix 1.) Associative A * ( B * C ) = ( A * B ) * multiplication C2.) Distributive A * ( B + C ) = ( A * B ) + ( A * C )(A + B ) * C = (A * C ) + ( B * C ) Linear Algebra: properties of Identity matrix If A is an identity matrix of order m x n thenA * I = AI * A = A Linear Algebra: transpose ( transpose( A ) ) A Linear Algebra: transpose( A + B ) transpose(A) + transpose(B) Linear Algebra: transpose( c * A) c * transpose(A) Linear Algebra: transpose ( A * B ) transpose(B) * transpose(A) Linear Algebra: skew symmetric matrix if matrix istranspose(A) = -A Linear Algebra: trace of n x n matrix Tr(A) = sum of main diagonal entriesa11 + a22 + ... + ann Linear Algebra: inverse of matrix n x n matrix A is invertible if there exists an n x n matrix such thatA * B = B * A = I (Identity matrix ) Linear Algebra: if matrix does not have an singular inverse it is ______ Linear Algebra: if matrix A is invertible( A^-1 A ) ^-1 Linear Algebra: if matrix A is invertible( A^k ) A^-1 * A^-1 * .... * A^-1k times ^ -1 = Linear Algebra: if matrix A is invertible( c * A 1/c * A^-1 ) ^ -1 Linear Algebra: if matrix A is invertible( transpose ( A ^ -1 ) transpose(A) ) ^-1 = Linear Algebra: inverse of a product of B^-1 * A^-1 matrices( A * B ) ^ -1 Linear Algebra: if C is an invertible matrix, 1.) A = B2. ) A = B then1.) if A*C = B*C then _____2.) if C*A = C * B then _____ Linear Algebra: if A is invertible matrix then X = A^-1 * B the solution ofA * X = B is Linear Algebra: what is an elementary matrix an nxn matrix that can be obtained from I^n by a single elementary row operation Linear Algebra: a square matrix A is it can be written as the product of elementary invertible if and only if _____ matrices Linear Algebra: define idempotent matrix square matrix A ifA^2 = A Linear Algebra: if A is a square matrix, then minor - the determinant of the matrix the minor Mij of the element aij is ___ and obtained by deleting the ith row and jth cofactor cij is ____ column of Acofactor - cij = (-1)^(i+j) Mij Linear Algebra: triangular matrix if all zero entries above or below its main diagonal Linear Algebra: if A is a triangular matrix of the product of the entries on the main order n, then its determinant is ___ diagonal| A | = a11 * a22 * .... * ann Linear Algebra: if A and B are square |A| * |B| matrices of order n then| A * B | = Linear Algebra: if A is an n x n matrix and c a c^n * |A| scalar then| c * A | = Linear Algebra: a square matrix A is not equal 0 invertible if and only if | A | = _____ Linear Algebra: if A is invertible, then | A ^ -1 1 / | A | | = Linear Algebra: an invertible square matrix A A ^ -1 = transpose( A ) is called orthogonal if ____ Linear Algebra: adjoint of matrix if A is a square matrix then matrix of cofactors =| C11 C12 .... C1n || C21 C22 ..... C2n|| Cn1 Cn2 .... Cnn|the transpose of this matrix is adjoint A Linear Algebra: relationship of inverse of nxn A ^ -1 = 1 / |A| * adj(A) matrix and its adjoint Linear Algebra: area of a triangle with 1/2 * determinant|x1 y1 1 ||x2 y2 1 ||x3 y3 1 | vertices (x1, y1), (x2, y2), (x3, y3) Linear Algebra: what information does means the transformation has an inverse existence of a determinant provide about a operation matrix that is a linear transformation of vector space Linear Algebra: geometric interpretation of * absolute value of determinant is scale determinant of square matrix with real factor by which area / volume is multiplied entries when being used as linear under linear transformation* sign indicates transformation of vector space whether transformation preserves orientation Linear Algebra: nilpotent matrix there exists a positive integer k such that A^k = 0 Linear Algebra: define subspace of vector subset W of a vector space V is a subspace space of V if W is itself a vector space under the operation of vector addition and scalar multiplication Linear Algebra: if V and W are both is also subspace of U subspaces of a vector space U, then the intersection of V and W ______ Linear Algebra: a vector v in a vector space V can be written in the formv = c1 u1 + c2 u2 V is called a linear combination of vectors + ... + ck ukwhere c1, c2, ..., ck are scalars u1, u2, ..., un in V if _____ Linear Algebra: define the spanning set of a S = { v1, v2, ..., vK) be a subset of vector vector space space V. The set S is called a spanning set of V if every vector of V can be written as a linear combination of vectors in S Linear Algebra: a set of vectors S = { v1, v2, the vector equationc1 v1 + c2 v2 + .... + ck ..., vn } is a vector space V is called linearly vk = 0has only the trivial solutionc1=0, c2=0, independent if _____ ... , ck = 0 Linear Algebra: distance between 2 vectors u || u - v || and v in R^n is Linear Algebra: the dot product ofu = ( u1, scalar quantityu dot v = u1*v1 + u2*v2 + .... + u2, ... un) andv = ( v1, v2, ...., vn) un*vn Linear Algebra: Cauchy-Schwarz Inequality If u and v are vectors in R^n, then| u dot v | <= ||u|| * ||v|| Linear Algebra: the angle theta between 2 cos(theta) = ( u dot v ) / ( ||u|| * ||v|| ) non zero vectors in R^n is given by Linear Algebra: 2 vectors u and v in R^n are u dot v = 0 orthogonal if ____ Linear Algebra: if u and v are vectors in R^n ||u||^2 + ||v||^2 then u and v are orthogonal if and only if ||u + v||^2 = _____ Linear Algebra: another name for dot product Euclidean inner product in R^n Linear Algebra: notation for dot product in dot - u . vgeneral - <u,v> R^n versus general inner product Linear Algebra: notation for dot product in dot - u . vgeneral - <u,v> R^n versus general inner product Linear Algebra: a vector space V with an an inner product space inner product is called _____ Linear Algebra: 4 axioms that define inner associate real number <u,v> with each pair product vector space V of vectors u + v1.) <u,v> = <v,u>2.) <u, v+w> = <u,v> + <u,w>3.) c*<u,v> = <c*u, v>4.) <v,v> >= 0 and <v,v> = 0 if and only if v=0 Linear Algebra: u is a vector in an inner ||u|| = square root( <u,u> ) product space Vnorm of u is ______ Linear Algebra: if u and v are vectors in inner || u - v || product space Vthe distance between u and v is ____ Linear Algebra: let u and v be vectors in an cos ( theta ) = <u,v> / ||u||*||v|| inner product space Vthe angle between 2 nonzero vectors u and v is _____ Linear Algebra: if u and v are vectors in an <u,v> = 0 inner product spaceu and v are orthogonal if Linear Algebra: if u and v are vectors in an ||u||^2 + ||v||^2 inner product space Vu and v are orthogonal if and only if ||u + v||^2 = _____ Linear Algebra: u and v are vectors in an (<u,v> / <v,v> ) * V inner product space Vthe orthogonal projection of u onto v is _____ Linear Algebra: orthogonal set of vectors S every pair of vectors in S is orthogonal in an inner product space V Linear Algebra: orthonormal set of vectors S every pair of vectors is orthogonal and each in an inner product space V vector is a unit vector Linear Algebra: coordinates for vector w w = <w,v1>v1 + <w, v2>v2 + ...