True/False Questions Linear Algebra PDF

111 TRUE/FALSE QUESTIONS Chapter 1: Systems of Linear Equations (1) A system of 3 linear equations in 2 unknowns must have no solution (2) A system of 2 linear equations in 3 unknowns could have exactly one solution (3) A system of linear equations could have exactly two solution...

111 TRUE/FALSE QUESTIONS Chapter 1: Systems of Linear Equations (1) A system of 3 linear equations in 2 unknowns must have no solution (2) A system of 2 linear equations in 3 unknowns could have exactly one solution (3) A system of linear equations could have exactly two solutions (4) If there’s a pivot in every row of A, then Ax = b is consistent for every b (5) If the augmented matrix has a pivot in the last column, then Ax = b is inconsistent (6) If A has a row of zeros, then Ax = b is inconsistent for all b (7) Ax = 0 is always consistent (8) If {u, v, w} is linearly dependent then {Au, Av, Aw} is also lin- early dependent for every A (9) If {u, v, w} is linearly independent and {v, w, p} is linearly inde- pendent, then so is {u, v, w, p} (10) If {u, v, w} is linearly dependent, then u is in the span of {v, w} (11) If {u, v, w} is linearly dependent and {u, v} is linearly indepen- dent, then w is in the span of {u, v} (12) If T is a linear transformation from R2 to R3 , then the matrix A of T is a 2 × 3 matrix Date: Thursday, March 14, 2019. 1 2 111 TRUE/FALSE QUESTIONS (13) If T (cu) = cT (u) for every real number c, then T is a linear trans- formation (14) If T (u + v) = T (u) + T (v) for all u and v, then T is a linear transformation (15) If T (u + cv) = T (u) + cT (v) for all u and v and every real number c, then T is a linear transformation (16) If T : R2 → R3 is linear, then T cannot be onto R3 (17) If T is one-to-one and {u, v, w} is linearly independent, then {T (u), T (v), T (w)} is also linearly independent Chapter 2: Matrix Algebra (18) AB + B T AT is always symmetric (19) Any matrix A can be written as a sum of a symmetric (AT = A) and antisymmetric (AT = −A) matrix (20) (AB)−1 = A−1 B −1 (21) If AB = AC, then B = C   1 2 3 (22) 2 4 6 is not invertible 3 6 9 (23) If AB = I for some B, then A is invertible (24) A 3 × 2 matrix could be invertible (25) A 2 × 3 matrix could be invertible (26) If AB is invertible, then A and B are invertible (27) Same, but this time A and B are square (28) If N ul(A) = {0}, then A is invertible  111 TRUE/FALSE QUESTIONS 3 (29) Every linear transformation T : Rn → Rm has a matrix (30) If T : Rn → Rn is one-to-one, then T is onto Rn (31) The row-operations that transform A to I also transform I to A−1 (32) If A is square and Ax = 0 implies x = 0, then A is row-equivalent to the identity matrix Chapter 3: Determinants (33) In general, det(2A) = 2 det(A) (34) det(A + B) = det(A) + det(B) (35) If det(A2 ) + 2 det(A) + det(I) = 0, then A is invertible (36) det(A−1 ) = − det(A) (37) If A100 is invertible, then A is invertible (38) If det(A) = 1 and A has only integer entries, then A−1 has integer entries (39) If det(A) = 1 and A and b have only integer entries, then the solu- tion x to Ax = b has only integer entries Chapter 4: Vector Spaces and Subspaces (40) {(x, y) ∈ R2 | x2 + y 2 = 0} is a subspace of R2 (41) The union of two subspaces of V is still a subspace of V (42) The intersection of two subspaces of V is still a subspace of V (43) Given any basis B of V , and a subspace W of V , then there is a subset of B that is a basis of W (44) R2 is a subspace of R3 4 111 TRUE/FALSE QUESTIONS     1 0 1 (45) N ul = Span 0 0 0 (46) For a fixed b 6= 0, the set of solutions to Ax = b is a subspace of Rn (47) If A is a 4 × 6 matrix with 2 pivot columns, then N ul(A) = R4 (48) If A is m × n and has n pivot columns, then N ul(A) = {0} (49) If A is m × n and has n pivot columns, then Col(A) = Rm (50) If A is row-equivalent to B, then Col(A) = Col(B) (51) Rank(A2 ) = Rank(A) (52) The set W of polynomials of degree n is a subspace of the set V of polynomials (of any degree) (53) If A is 5 × 9, then N ul(A) is at least 4 dimensional (54) cos2 (t), sin2 (t), cos(2t) is linearly dependent  (55) Z is a subspace of R (56) If W is a subset of V such that 0 (the zero vector in V ) is in W and W is closed under addition, then W is a subspace of V (57) If 0 is in W and W is closed under scalar multiplication, then W is a subspace of V (58) If W is closed under addition and scalar multiplication, then W is a subspace of V (59) A vector space V is always a subspace of something (60) If A s row-equivalent to B, then the pivot columns of B form a basis for Col(A) (61) Row-operations preserve the span of the columns of a matrix  111 TRUE/FALSE QUESTIONS 5 (62) Row-operations preserve the linear independence relations of the columns of a matrix (63) If B spans a space V , then there is a subset of B that is a basis for V (64) If B = {v1 , · · · , vn } is a linearly independent subset of a n−dimensional vector space V , then B is a basis for V (65) If B = {v1 , · · · , vn } is a spanning subset of a n−dimensional vec- tor space V , then B is a basis for V (66) dim(P4 ) = 4 (67) If B is a basis for Rn and P is a matrix with the vectors of B as its columns, then P x = [x]B (the coordinates of x with respect to B)   (68) If B = {b1 , · · · , bn } and C are bases of Rn and P = [b1 ]C · · · [bn ]C , then P [x]C = [x]B (69) If B = {e1 , · · · , en } is the standard basis of Rn , then [x]B = x (70) Rank(A) = Rank(AT ) Chapter 5: Eigenvalues and Eigenvectors (71) A 3 × 3 matrix with eigenvalues λ = 1, 2, 4 must be diagonalizable (72) A 3 × 3 matrix with eigenvalues λ = 1, 1, 2 is never diagonalizable (73) Every matrix is diagonalizable (74) If A is similar to B, then det(A) = det(B) (75) If A is similar to B, then A and B have the same eigenvalues (76) If A is diagonalizable, then det(A) is the product of the eigenvalues of A (77) If A is similar to B, then A and B have the same eigenvectors 6 111 TRUE/FALSE QUESTIONS (78) If A is invertible, then A is diagonalizable (79) If A is diagonalizable, then A is invertible (80) If A is similar to B, then A2 is similar to B 2 (81) If A is diagonalizable and invertible, then A−1 is diagonalizable (82) If λ = 0 is an eigenvalue of A, then A is not invertible (83) (Nonzero) Eigenvectors corresponding to different eigenvalues of A are linearly independent (84) Every matrix has a real eigenvalue (85) Every matrix has a complex eigenvalue (86) If the characteristic polymomial of A is λ2 − 3λ + 2 = 0, then A2 − 3A + 2I = O (the zero-matrix) Chapter 6: Orthogonality and Least-Squares (87) If x̂ is the orthogonal projection of x on a subspace W , then x̂ is perpendicular to x ˆ = x̂ (88) x̂ (89) The orthogonal projection of x on W ⊥ is x − x̂ (90) Every (nonzero) subspace W has an orthonormal basis (91) W ∩ W ⊥ = {0} (92) AAT x is the projection of x on Col(A) (93) Same, but the columns of A are orthonormal (94) Rank(AT A) = Rank(A) (95) If Q is an orthogonal matrix, then Q is invertible  111 TRUE/FALSE QUESTIONS 7 (96) If Q is a matrix with orthonormal columns, then kQxk = kxk (97) An orthogonal set without the zero-vector is linearly independent u·v
(98) The orthogonal projection of v on W = Span {u} is v·v v (99) An orthogonal matrix has orthogonal columns (100) If x̂ is a least-squares solution of Ax = b, then x̂ is the orthogonal projection of x on Col(A). (101) If x̂ is a least-squares solution of Ax = b, then Ax̂ is the point on Col(A) that is closest to b (102) Ax = b has only one least-squares solution (103) If ku + vk2 = kuk2 + kvk2 , then u is orthogonal to v (assume that everything is real) R1 R  12 R  21 1 1 (104) 0 f (x)g(x)dx ≤ 0 (f (x))2 dx 0 (g(x))2 dx (105) The product of two orthogonal matrices (it it’s defined) is orthogo- nal (106) Col(A) is orthogonal to N ul(AT ) Chapter 7: Symmetric Matrices (107) If A is symmetric, then eigenvectors corresponding to different eigen- values of A are orthogonal (108) A symmetric matrix has only real eigenvalues (109) Linearly independent eigenvectors of a symmetric matrix are or- thogonal (110) If A is symmetric, then it is orthogonally diagonalizable (111) If A is orthogonally diagonalizable, then it is symmetric