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Category: Default - (148 questions) Linear Algebra: 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 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 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 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 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 if matrix istranspose(A) = -A Linear Algebra: 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 , 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 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 square matrix A ifA^2 = A Linear Algebra: if A is a square matrix, then - 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: 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: 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 - R^n versus general inner product Linear Algebra: notation for dot product in dot - u . vgeneral - 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 with each pair product vector space V of vectors u + v1.) = 2.) = + 3.) c* = 4.) >= 0 and = 0 if and only if v=0 Linear Algebra: u is a vector in an inner ||u|| = square root( ) 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|| inner product space Vthe angle between 2 nonzero vectors u and v is _____ Linear Algebra: if u and v are vectors in an = 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 ( / ) * 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 = v1 + v2 + .... vn relative to orthonormal basis B = {v1, v2, ..., vn } with respect to inner product space V Linear Algebra: Gram-Schmidt 1.) B' = {w, w2, ..., wn} where wi is given Orthonormalization process for B = {v1, v2, byw1 = v1w2 = v2 - / * ..., vn) as basis for inner product space V w1w3 = v3 - /*w2.....2.) ui = wi / ||wi||, then B'' = { u1, u2, ..., un} Linear Algebra: cross product of 2 vectors in u = u1 * i + u2 * j + u3 *kv = v1 * i + v2 *j + v3 R^3u x v *ku x v = (u2v3 - u3v2) i - (u1v3 - u3v1) j + (u1v2 - u2v1)k Linear Algebra: if u and v are nonzero orthogonal ( to both u and v ) vectors in R^3u X v is ______Linear Algebra: if u and v are nonzero || u x v || = ||u|| * ||v|| sin(theta) vectors in R3 then the angle between u and v = Linear Algebra: let f be continuous on [a,b] the value ofI = integral over [a,b] ( f(x) - g(x) and let W be a subspace of C[a,b].A function )^2 dxis a minimum with respect to all other g in W is called least squares approximation function in W of f with respect to W if _____ Linear Algebra: linear transformation V and W are vector spacesfunction T: V -> W is linear transformation of V into W if the following 2 properties are true for all u and v in V and for any scalar c1.) T(u + v) = T(u) + T(v)2.) T(c*u) = c * T(u) Linear Algebra: linear operator a linear transformation T:V->V from a vector space into itself Linear Algebra: kernel of a linear Let T: V -> V be a linear transformationThen transformation the set of all vectors v in V that statisfy T(v) = 0 is called the kernel of T and denoted byker(T) Linear Algebra: the range of linear subspace transformation T : V -> W is _____ of W Linear Algebra: if T: V->W is a linear the dimension of the kernel of T transformation - define the nullity of T Linear Algebra: if T:V->W is a linear dimension of the range of T transformationdefine the rank of T Linear Algebra: let T:V->W be a linear rank + nullity = ndim(range) + dim(kernel) = transformation from n-dimensional vector dim(domain) space V into a vector space WRelationship of rank / nullity / n Linear Algebra: define one-to-one linear if the preimage of every W consists of a transformationT: V->W single vector Linear Algebra: what kernel implies that kernel(T) = { 0 } linear transformation is one to one Linear Algebra: a function T:V->W is said to every element in W has a preimage in be onto if _____ VWhen W is equal to the range of T Linear Algebra: define isomorphism linear it is one-to-one and onto transformation Linear Algebra: isomorphic vectors V and W if there exists an isomorphism from V to W(one to one and onto) Linear Algebra: condition necessary for 2 if they have the same dimension vectors to be isomorphic Linear Algebra: the composition, T, of T(v)=T2(T1(V)) T1:R^n->R^m with T2:R^m->R^p is defined by ____ Linear Algebra: matrix formula for A = A2*A1 composition of T(v)=T2(T1(v) where A1, A2 are standard matrices for T1, T2 Linear Algebra: let T:R^n->R^n be a linear column space of A transformation given by T(x) = Ax. Then the ___ of A is equal to the range of T Linear algebra: linear transformation T(v) = A[V]B = [T(v)]B' T:V->W, where B and B' are basis for V and W.Matrix of T relative to bases B and B' Linear algebra: how to find the linear [T(v1)] = [a11, a21, am1], ..... [T(vn)] = [a1n, for nonstandard Basis a2n,....,amn]then the mxn matrix| a11 ..... B={v1, v2, .... vn} and B'T: V->W a1n || a21 ..... a2n || ...... ||am1...... amn| Linear algebra: definition of similar matrices for square matrices A and A' of order n,A' is said to be similar to A if there exists an invertible matrix P such thatA' = P^-1 * A * P Linear algebra: property of similar similar matricesLet A be a square matrix of order n, A is ____ to A. Linear algebra: let A, B be square matrices similar to A of order n:if A is similar to B, then B is ____ Linear algebra: let A, B, and C be square A is similar to C matrices of order n.If A is similar to B and B is similar to C, then Linear algebra: matrix for reflection in y-Axis A =| -1 0|| 0 1 | in 2D plane Linear algebra: matrix for linear A =| 1 0 || 0 -1 | transformation in plane, reflection in X-axis Linear algebra: matrix for reflection in line y A =| 0 1 || 1 0 | = x Linear algebra: matrix for horizontal A =| k 0 || 0 1 | expansion / contraction in 2D Linear algebra: matrix for vertical expansion | 1 0 || 0 k | / contraction in 2D Linear algebra: matrix for horizontal shear in | 1 k || 0 1 | 2D Linear algebra: matrix for vertical shear in | 1 0 || k 1 | 2D Linear algebra: eigenvalue of n x n matrix A the scalar b is called eigenvalue of A if there is a nonzero vector x such thatAx = bx Linear algebra: eigenvector of n x n matrix A if there is a nonzero vector x such thatAx = bx with scalar b Linear algebra: eigenspace of b for n x n the set of all eigenvectors of b together with matrix A with eigenvalue b zero vector is subspace of R^n Linear algebra: what equation can be used scalar b such thatdet( b * I - A) = 0I - is n x n to find eigenvalues of n x n matrix A identity matrix Linear algebra: equation for finding nonzero solutions of( b I - A)x = 0 b - eigenvectors of n x n matrix A eigenvalue I - n x n identity matrix Linear algebra: characteristic equation of n x det( b * I - A ) = 0 n matrix A Linear algebra: characteristic polynomial of | b * I - A | = b^n + c*b^n-1 + ..... c1*b + C0 n x n matrix A Linear algebra: eigenvalue problem if A is an n x n matrix, are there nonzero vectors x in R^n such that Ax is a scalar multiple of x Linear Algebra: define a diagonalizable a n x n matrix A is diagonalizable if A is matrix similar to a :if there exists an invertible matrix P such that: P^-1 * A * P is a diagonal matrix Linear Algebra: _____ matrices have the similar same eigenvalues Linear Algebra: a n x n matrix is it has n linearly independent eigenvectors diagonalizable if and only if _____ Linear Algebra: definition of symmetric A = transpose ( A ) matrix Linear Algebra: if A is an n x n symmetric 1. A is diagonalizable2. All eigenvalues of A matrix then the following 3 properties are are real.3. If b is an eigenvalue of A with true multiplicity k, then b has k linearly independent eigenvectors. The eigenspace of b has dimension k. Linear Algebra: definition of orthogonal a square matrix PP^-1 = transpose of P matrix Linear Algebra: if A is n x n symmetric and b1 and b2 are distinct eigenvalues of A, then their corresponding eigenvectors x1 and x2 are ______Linear Algebra: fundamental theorem of Let A be a n x n matrix, then A is symmetric matrices orthogonally diagonalizable if and only if A is symmetric Linear Algebra: define multiplicity of if eigenvalue b occurs as a multiple root k eigenvalue times for the characteristic polynomial Linear Algebra: the main property of a it reduces dimensionality projection is ______Linear Algebra: eigenvectors are vectors parallel to xAx = lambda * xlambda = multiple that when multiplied by matrix A come out ______Linear Algebra: if A is singular, then lambda 0 = _____ is an eigenvalue Linear Algebra: sum of the eigenvalues of trace of A matrix A = ______Linear Algebra: the determinant of singular 0 matrix = ______Linear Algebra: the determinant of singular 0 matrix = ______Linear Algebra: formula to map (u1, u2) in x1 = (1 - u1)*min1 + u1 * max1x2 = (1 - local coordinates to global coordinates with u2)*min2 + u2 * max2 target box (min1, min2) and (max1, max2) Linear Algebra: barycentric combination sum of points where the coefficients sum to 1, which the location of a point is specifies as the center of the mass of other points Linear Algebra: a unit vector that is normal perpendicular to a line is referred to as the _____ of the line Linear Algebra: 2 parametric equation of a parameter t1.) L(t) = p + t * v2.) L(t) = (1-t) p line + t q Linear Algebra: implicit equation of the line a*x1 + b*x2 + c = 0x1 - x coordinatex2 = y coordinate Linear Algebra: those transformations that rigid body motions are described by orthogonal matrices are called ______Linear Algebra: the determinant of an +/- 1 orthogonal matrix is ______Linear Algebra: a system of the form Au = 0 homogeneous is called ______Linear Algebra: affine map x' = Ax + p Linear Algebra: if an object is moved without translated changing the orientation then it is _____ Linear Algebra: equation for translation in x' = p + IxI - identity matrix 2D Linear Algebra: physical meaning of it is a vector that when multiplied against A eigenvectors of matrix A expands or contracts in same direction as vectorA * eigenvector = lambda * eigenvector Linear Algebra: physical meaning of the amount the eigenvector shrinks / grows eigenvalues of matrix A when multiplied by AA * x = eigenvalue * x Linear Algebra: eigenvector corresponding kernel of the matrix to the zero eigenvalue is ______Linear Algebra: A 2D projection matrix 0 and 1 always has eigenvalues of ___ and ____