2. Every linear system with the same number of equations as unknowns has a unique solution.


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1 1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations as unknowns has at least one solution. 4. Every linear system with more equations than unknowns may have an infinite number of solutions. 5. Every linear system with fewer equations than unknowns may have no solutions. 6. Every matrix is row equivalent to a unique matrix in row echelon form. 7. If [A b] and [C d] are row equivalent augmented matrices, the matrix equations Ax = b and Cx = d have the same solution set. 8. A linear system with a square coefficient matrix A has a unique solution if and only if A is row equivalent to the identity matrix. 9. A linear system with coefficient matrix A has an infinite number of solutions if and only if A can be row reduced to an echelon form that includes some column containing no pivot. 10. A consistent linear system with coefficient matrix A has an infinite number of solutions if and only if A can be row reduced to an echelon form that includes some column containing no pivot. 11. If a square linear system Ax = b has a solution for every choice of column vector b, then the solution is unique for each b. 12. If a square linear system Ax = 0 has only the trivial solution, then Ax = b has a unique solution for every (appropriately sized) column vector b. 13. Multiplication of a nonzero vector by a nonzero scalar never yields the zero vector 14. No vector is its own additive inverse. 15. Every vector space has at least two vectors. 16. Every vector space has at least two distinct subspaces. 17. If u + v is in a subspace W of a vector space V, then both u and v are elements of W. 18. Two subspaces of a vector space V may have an empty intersection.
2 19. The rank of A + B is less than or equal to the ranks of both A and B. 20. Let A and B be 2 2 matrices such that AB = 0. Either A = 0 or B = 0, or both. 21. Let A and B be 2 2 matrices such that AB = 0. If B is invertible, then A = Let A and B be 2 2 matrices such that AB = 0. BA = Let A and B be 2 2 matrices such that AB = 0. There is a vector x 0 such that BAx = If B is an n n matrix and B 2 = B, then B is not invertible. 25. The span of any two nonzero vectors in R 2 is all of R If v 1, v 2,..., v k are vectors in R 2 such that span(v 1, v 2,..., v k ) = R 2, then k = A subset of R n containing two nonzero distinct parallel vectors is dependent. 28. If a set of nonzero vectors in R n is dependent, then any two vectors in the set are parallel. 29. Every subset of three vectors in R 2 is dependent. 30. Every subset of two vectors in R 2 is independent. 31. If a subset of two vectors in R 2 spans R 2, then the subset is independent. 32. If S is independent, then each vector in V can be expressed uniquely as a linear combination of the vectors in S. 33. If S is independent and spans V, then each vector in V can be expressed uniquely as a linear combination of the vectors in S. 34. Every independent subset of V is a subset of some basis for V. 35. If H is a row echelon form of a matrix A, then the nonzero column vectors in H form a basis for the column space of A. 36. For all positive integers m and n, the nullity (dimension of the null space) of an m n matrix might be any number from 0 to n. 37. For all positive integers m and n, the nullity of an m n matrix might be any number from 0 to m.
3 38. For all positive integers m n, the nullity of an m n matrix might be any number from 0 to n. 39. There is a unique coordinate vector associated with each vector v V. 40. There is a unique coordinate vector associated with each vector v V relative to a basis for V. 41. There is a unique coordinate vector associated with each vector v V relative to an ordered basis of V. 42. Distinct vectors in V have distinct coordinate vectors relative to the same ordered basis B of V. 43. The same vector in V cannot have the same coordinate vector relative to different ordered bases B and C of V. 44. Every changeofcoordinates matrix is square. 45. There are six possible ordered bases for R There are six possible ordered bases for R 3 consisting of the standard basis vectors. 47. If A is invertible, the row space of A 1 must be the same as the row space of A. 48. If A is a 6 8 matrix then the dimension of Null(A) is at least two. 49. If A is a 6 8 matrix and the dimension of the null space of A is three, then the dimension of Col(A T ) is five. 50. If A is a 3 2 matrix with independent columns, then there exists a 2 3 matrix B such that AB = I Let A be an n n matrix. If B = {b 1, b 2,..., b n } is a basis for the row space of A (these b i s are written in row b 1 vector shape), then the columns of the matrix b 2. b n form a basis for the column space of A. 52. Let V and W be linear spaces (vector spaces) and let T : V W be a linear transformation. Let v 1, v 2..., v n be vectors in V. If T(v 1 ), T(v 2 ),..., T(v n ) are linearly independent, then v 1, v 2..., v n are linearly independent. 53. The determinant of a 2 2 matrix is a vector. 54. If two rows of 3 3 matrix are interchanged, the sign of the determinant is changed. 55. The determinant of a 3 3 matrix is zero if two rows of the matrix are parallel vectors in R 3
4 56. In order for the determinant of a 3 3 matrix to be zero, two rows of the matrix must be parallel vectors in R The determinant of A, det(a), is defined for any matrix A. 58. det(a) is defined for each square matrix A. 59. det(a) is a scalar. 60. If a matrix A is multiplied by the scalar k, then the determinant of the resulting matrix is k det(a). 61. If an n n matrix A is multiplied by the scalar k, then the determinant of the resulting matrix is k det(a). 62. If an n n matrix A is multiplied by the scalar k, then the determinant of the resulting matrix is k n det(a). 63. det(aa T ) = det(a T A) = [det(a)] The determinant of an elementary matrix is nonzero. 65. If det(a) = 2 and det(b) = 2, then det(a + B) = If det(a) = 2 and det(b) = 2, then det(ab) = If A and B are n n matrices and det(a) = 2 and det(b) = 3, then det(ab) = The determinant of a square matrix is the product of the entries on its main diagonal. 69. The determinant of an upper triangular matrix is the product of the entries on its main diagonal. 70. The determinant of a lower triangular matrix is the product of the entries on its main diagonal. 71. A square matrix is invertible if and only if its determinant is positive. 72. The column vectors of an n n matrix are independent if and only if the determinant of the matrix is nonzero. 73. A homogeneous square linear system has a nontrivial solution if and only if the determinant of its coefficient matrix is zero. 74. Every n n matrix has real eigenvalues. 75. There can only be one eigenvalue associated with an eigenvector.
5 76. There can only be one eigenvector associated with a distinct eigenvalue. 77. If v is an eigenvector for A then v is an eigenvector for A ki n, for all scalars k. 78. If ł is an eigenvalue for A then ł is an eigenvalue for A ki n, for all scalars k. 79. If v is an eigenvector for A 1 then kv is an eigenvector for A, for all nonzero scalars k. 80. Every vector in R n is an eigenvector for I n. 81. If an n n matrix has n distinct real eigenvalues it is diagonalizable. 82. An n n matrix is diagonalizable if and only if it has n distinct eigenvalues. 83. Let A and B be 3 3 matrices. If det(a) = 1 and det(b) = 0, then rank A > rank B. 84. An n n matrix is diagonalizable if and only if the algebraic multiplicity of each of its eigenvalues equals the geometric multiplicity. 85. Every invertible matrix is diagonalizable. 86. Every triangular matrix is diagonalizable. 87. If A and B are similar and A is diagonalizable, then B is also diagonalizable. 88. If A and B are similar, then det(a) = det(b). 89. If A is diagonalizable, there is a unique diagonal matrix D that is similar to A. 90. If A and P are n n matrices with P invertible, then det(pap 1 ) = det(a). 91. Similar matrices have the same eigenvalues and eigenvectors. 92. Similar matrices have the same eigenvalues with the algebraic and geometric multiplicities. 93. If A and B are n n matrices with B invertible and v an eigenvector of A, then Bv is an eigenvector of BAB If A and B are n n matrices with B invertible and v an eigenvector of A, then B 1 v is an eigenvector of B 1 AB.
6 95. Any two n n diagonalizable matrices having the same eigenvalues of the same algebraic multiplicities are similar. 96. Any two n n diagonalizable matrices having the same eigenvectors are similar. 97. Any two n n diagonal matrices are similar. 98. Every nonzero vector in R n has nonzero magnitude. 99. Every vector of nonzero magnitude in R n is nonzero There are exactly two unit vectors parallel to any given nonzero vector in R n There are exactly two unit vectors orthogonal to any given nonzero vector in R n The dot product of a vector with itself yields the magnitude of the vector For a vector v R n and r a scalar, the magnitude of r times v is r times the magnitude of v If v and w are vectors in R n of the same magnitude, then the magnitude of v w is The set of all vectors in R n orthogonal to every vector w of a subspace W is a subspace of R n The intersection of W and W is empty If vectors u and v have the same projection onto the subspace W of V, then u = v All vectors in an orthogonal basis have length If A has kernel {0}, then the Gram matrix, A T A, is invertible The parallelogram in R 2 determined by nonzero vectors x and y is a square if and only if x y = The box in R 3 determined by vectors x, y and z is a cube if and only if x y = x z = y z0, and x x = y y = z z The projection of x onto the span of y is a scalar multiple of x The projection of x onto the span of y is a scalar multiple of y The set of all vectors in R n which are orthogonal to a subspace W of R n is a subspace of R n.
7 115. If the projection of x onto the subspace W of R n is x itself, then x is orthogonal to every vector in W If the projection of x onto the subspace W of R n is x itself, then x W The intersection of W and W is empty if proj W x = proj W y then x = y All vectors in an orthogonal basis have length Every nontrivial subspace of R n has an orthonormal basis Every vector in R n is in some orthonormal basis for R n Every unit vector in R n is in some orthonormal basis for R n A matrix is orthogonal when its column vectors are orthogonal A square matrix is orthonormal when its column vectors are orthonormal Every orthogonal matrix has a trivial nullspace If A T is orthogonal then A is orthogonal If A is a symmetric orthogonal n n matrix, then A 2 = I n If A is a symmetric n n matrix with A 2 = I n, then A is an orthogonal matrix If A and B are both n n orthogonal matrices, then AB is an orthogonal matrix.
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