lin alg final t/f

5.3 All nonzero symmetric matrices are invertible

False consider [1 1 ;1 1]

8.1 if A is an orthogonal matrix then there must exist a symmetric invertible matrix S such that S^-1AS is diagonal

False, A being orthogonal does not mean it has to be diagonalizeable

5.1 if A and B are symmetric nxn matrices then A + B must be symmetric as well

True (A+B)^t = A^t + B^t = A + B so A +B )^t = A +B so it is symmetric

5.2 if matrices A and S are orthogonal then S^-1AS is orthogonal as well

True S^-1 is orthogonal and the product of two orthogonal matrices is orthogonal

5.14 if A is any matrix with kerA = 0 then the matrix AA^t represents the orthogonal projection onto the image of A

false

5.21 If A and B are orthogonal 2x2 matrices then AB = BA

false

7,37 if v and w are linearly independent eigenvectors of matrix A, then v+w must also be an e-vector of A

false

7.24 if two nxn matrices A and B are diagonizeable, then A + B also must be diagonizeable

false

7.25 all diagonalizeable matrices are invertible

false

7.30 if two nxn matrices A and B are diagonalizeable then AB must be as well

false

7.36 all invertible matrices are diagonizeable

false

7.41 all orthogonal matrices are diagonalizeable over R

false

7.52 if two nxn matrices A and B are both diagonializeable then they must commute

false

8.21 if v and w are lin independent e vectors of a symmetric A, then w must be orthogonal to v

false

8.20if A is any matrix then matrix A^tA is the transpose of AA^t

false (AA^t)^t = AA^t

5.12 if A and B are symmetric nxn matrices then AB must be symmetric as well

false (AB)^t = B^tA^t = BA =/= AB

5.37 if matrix A is similar to B and A is orthogonal then B is orthogonal as well

false B is orthogonal if P is

5.33 if A is an invertible matrix such that A^-1 = A then A must be orthogonal

false A^2 = I this is not necessarily orthogonal

5.29 [3 -4;4 3] is an orthogonal matrix

false columns are not unit vectors

5.13 if matrices A and B commute then A must commute with B^t as well

false consider A = B = [0 1;0 0]

7.29 all lower triangular matrices are diagonalizeable over C

false consider A = [0 0;2 0]

7.27 if matrix A^2 is diagonalizeable, then matrix A must be diagonizeable as well

false consider A = [0 2;0 0]

7.20 if n real matrix A has only the eigenvalues 1 and -1, A must be orthogonal

false consider [-1 1;0 1]

5.25 the determinant of all orthogonal 2x2 matrices is 1

false consider [1 0;0 -1]

5.17 there exists a subspace V of R^5 st dimV = dimV** where V** is the orthogonal complement of V

false dimV + dimV** = dimension of the space, since V*** + V forms a basis of the space. Thus dimV cannot = dimV** as one is even and one is odd

7.6 there exists a real 5x5 matrix without any real e-values

false each imaginary e value comes with a partner, so even if there are two, one has to be real

5.5 if u is a unit vector in R^n and L = span u then projL(x) = (x•u)x for all vectors x in R^n

false it is (x•u)u no bottom part because the projection is onto a unit vector

7.1 the algebraic multiplicity of an eianvalue cannot exceed its geometric multiplicity

false it is opposite algmult ≥ geomult

7.33 if an nxn matrix A is diagonalizeable, then A must have n distinct e values

false the geo mult must add to n, but the number of e values could be smaller

7.22 if detA = detA^t then A must be symmetric

false this equation is ALWAYS true

7.48 if a matrix A has k distinct e values then rankA >= k

false you can have k distinct e values but not have an invertible matrix, so there are not k pivots

5.10 the equation (AB)^t = A^tB^t holds for all nxn matrices A and B

false (AB)^t = B^tA^t

7.15 if 1 is the only e-value of an nxn matrix A then A must be Identity matrix

false consider [1 1;0 1]

7.35 if v is an e vector of A then v must be an e vector of A^t as well

false consider '0 1;0 1]

7.16 if A and B are nxn matrices, if a is an e value of A and b is an e value of B, ab must be an e-value of AB

false, if it was for one e vector v this may be true but not for the same e value you can't say

5.7 if T is a linear transformation from R^n to R^n such that T(e1) - T(en) are unit vectors, T is an orthogonal transformation

false, just because norm is preserved for e1-en does not mean it always is

7.14 if A and B are two 3x3 matrices susi that trA = trB and detA = detB, then A and B have the same e values

false, the equation only works for 2x2

7.43 if two matrices A and B have the same characteristic polynomials they must be similar

false, they must also have the same multiplicities

5.19if x and y are two vectors in R^n then the equation ||x+y||^2 = ||x||^2 + ||y||^2 must hold

false. this only holds if they are orthogonal

5.24 if A^tA = AA^t for an nxn matrix A then A is orthogonal

false?

5.20 the equation det(A^t) = det(A) holds for all 2x2 matrices

true

5.30 if V is a subspace of R^n and x is a vector in R^n then projv(x) must be orthogonal to vector x - projv(x)

true

5.31 there exist orthogonal 2x2 matrices A and B such that A+B is orthogonal as well

true

5.34 if the entries of two vectors v and w in R^n are all positive then v and w must enclose an acute angle

true

5.39 if matrix A is symmetric and matrix S is orthogonal, S^-1AS must be symmetric

true

5.44 if V is a subspace of R^n and x is a vector in R^n then the inequality x-projv(x)>=0 must hold

true

5.8 if A is an invertible matrix then the equation (A^t)^-1 = (A^-1)^t must hold

true

7.10 the trace of any square matrix is the sum of its diagonal entries

true

7.11 any rotation dilation matrix in R^2x2 is diagonizeable over C

true

7.17 if 3 is an e-value of an nxn matrix A then 9 must be an e-value of A^2

true

7.18 The matrix of any orthogonal projection onto a subspace of V of R^n is diagonizeable (this is the matrix that would be the transformation T(x) = projv(x))

true

7.2 if an nxn matrix A is diagonalizeable over R then there must be a basis fo R^n consisting of eigenvectors of A

true

7.21 if an invertible matrix A is diagonilizeable, then A^-1 must be diagonizeable as well

true

7.28 the determinant of a matrix is the product of its e-values over C counted with their algebraic multiplicities

true

7.3 if the standard vectors e1-en are e vectors of an nxn matrix, A must be diagonal

true

7.31 if u, v, w are eigenvectors of a 4x4 matrix A with associated e values 3, 7, and 11, then u, v, w are lin indep

true

7.39 if A is a 2x2 matrix such that trA = 1 and detA = -6 then A is diagonalizeable

true

7.40 if a matrix is diagonalizeable then the algebraic multiplicity of each of its e values must equal the geo mulitplicity of its e value

true

7.44 if A is a diagonalizeable 4x4 matrix with A^4 = 0 then A must be 0 matrix

true

7.47 similar matrices have the same characteristic poly

true

7.49if the rank of a square matrix is 1 then all nonzero vectors in the image of A are e vectors of A

true

7.5 there exists a diagonizeable 5x5 matrix with only two distinct eigenvalues over C

true

7.53 if v is an e vector of A then v must be in the kernel of A or the image of A

true

7.54All symmetric 2x2 matrices are diagonalizeable over R

true

7.55 if A is a 2x2 matrix with e values 3 and 4 and if u is a unit e vector of A then the length of vector Au cannot exceed 4

true

7.56 if u is a nonzero vector in R^n then u must be an e vector of matrix uu^t

true

7.57 if v1 - vn is an e basis for both A and B then matrices A and B must commute

true

7.8 The eigenvalues of a 2x2 matrix A are the solutions of the equation y^2 + trAy + detA = 0

true

7.9 the eigenvalues of any triangular matrix are its diagonal entries

true

8.13 if A is a symmetric matrix such that Av = 3v and Aw = 4w then the equation <v, w> = 0 must hold

true

8.16if A is any matrix then AA^t is diagonalizeable

true

8.18matrix [3 2 1;2 3 2; 1 2 3] is diagonalizeable

true

8.9 all symmetric matrices are diagonalizeable

true

5.36 the matrix A^tA is symmetric for all matrices A

true A^tA = (A^tA)^t = A^tA

5.23 the formula kerA = ker(A^tA) hold for all A

true if Ax = 0 A^tAx = A^t0 A^t0 = A^t0 = 0

5.46 if A is any symmetric 2x2 matrix there must exist real number x such that A-xI fails to be invertible

true take the determinant and see that there exist an x st it = 0

5.6 if A is symmetric matrix then 7A must be symmetric as well

true (7A)^T = 7A^t = 7A

5.15 if A and B are symmetric nxn then ABBA must be symmetric as well

true (ABBA)^t = (BA)^t(AB)^t = A^tB^tB^tA^t = ABBA

5.16 if matrices A and B commute, then A^t and B^t commute as well

true A^tB^t = (BA)^t = (AB)^t = B^tA^t

7.51 if A is 4x4 matrix with A^4 = 0 then 0 is the only e value of A

true Av = yv A^4v = 0v = y^4v if v is not 0, y must be 0

5.9 if matrix A is orthogonal then A^2 must also be orthogonal

true an orthogonal matrix times an orthogonal matrix is orthogonal. this is the definition

5.28 every non zero subspace of R^n has an orthonormal basis

true find with gramm schmidt

5.47 there exists a basis of R^2x2 that consists of orthogonal matrices

true gram shmit

7.7 if 0 is an e-value of matrix A then detA = 0

true if 0 is an e value, that means some vector which is the eigen vector is in the kernel despite not being 0. thus det = 0

5.40 if A is a square matrix such that A^tA = AA^t then kerA = kerA^t

true if A is orthogonal it is invertible so both kernels are 0

5.22 if A is a symmetric matrix vector v is in the image of A and vector w is the kernel of A, <v, w> = 0 must hold

true if w is in the kernel of A then Aw = 0 thus rows of A are perpendicular to w. since A is symmetric, the fact that rows are perpendicular to w means columns also are

5.45 if A is an nxn matrix such that ||Au||=1 for all unit vectors u then A must be orthogonal

true length is preserved, this is on the list

7.12 if A is nonintervtible nxn matrix, then the geo multiplicity of e - value 0 is n - rank(A)

true n - rank(A) is exactly the dimension of the kernel space, which is all the vectors that are e vectors for e value 0

7.13 if matrix A is diagonizeable then its transpose A^t must be diagonalizable as well

true A and A^t have the same characteristic poly so they have the same e values AND the same multiplicities and thus the same diagonalizeability

7.46 if vector v is an eigenvector of both A and B then v is an e vector of AB

true Av = av Bv = bv ABv = Abv = bAv = bav

7.38 if 2x2 matrix R represents a reflection about the line L, then R is diagonalizeable

true it is an o normal e basis as it preserves length

5.11 if matrix A is orthogonal then A^t must be orthogonal as well

true, A^t = A^-1, which is orthogonal so A^t is orthogonal

5.27 the entries of an orthogonal matrix are all less than or equal to 1

true, all vectors in the matrix are orthonormal

7.26 if vector v is an eigenvector of both A and B then v must be an e-vector of A + B

true, e value is a+b where a is e value of A and b is e value of B

7.32 if a 4x4 matrix A is diagonalizeable, then the matrix A + 4I must also be diagonalizeable as well

true, e value is y + 4 with y is the e value of A. so the e bases are the same and those diagonalizeability are the same

7.45 if an nxn matrix A is diagonalizeable over R then every vector v in R^n can be expressed as a sum of eigenvectors of A

true, there is an e basis

5.4 If A is an nxn matriculates such that AA^t = I then A must be an orthogonal matrix

true, this is the definition of an orthogonal matrix, that A^t = A^-1

7.4 if v is an e-vector of A then v must be an e vector of A^3 as well

true, with e value y^3

7.34 if two 3x3 matrices A and B both have e values 1, 2, 3, A and B must be similar

true. they have the same e values so they have the same diagonal D so they are similar

7.23 if matrix A = [7 a b;0 7 c;0 0 7] is diagonizeable, then a, b, c, must = 0

true?