Midterm Questions From Class Flashcards
T or F. Every elementary row operation is reversible
True
T or F. A 5x6 matrix has six rows.
False
T or F. The solution set of a linear system involving variables x1…xn is a list of numbers (s1,…sn) that makes each equations in the system a true statement when the values s1,…sn are substituted for x1,…xn respectively
False
T or F. Two fundamental questions about a linear system involve existence and uniqueness.
True
Elementary row operations on an augmented matrix never change the solution set of the associated system.
True
T or F. Two matrices are row equivalent if they have the same number if rows
False
T or F. An inconsistent system has more than one solution
False
T or F. Two linear systems are equivalent if they have the same solution set
True
T or F. In some cases a matrix may be row reduced to more than one matrix in reduced echelon form using different sequences of row operations.
False
T or F. In some cases a matrix may be row reduced to more than one matrix in reduced echelon form using different sequences of row operations.
False
T or F. The row reduction algorithm applies only to augmented matrices for a linear system.
False
T or F. A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix
True
T or F. Finding a parametric description of the solution set of a linear system is the same as solving the system.
True
T or F. If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent
False
T or F. The echelon form of a matrix is unique
False
T or F. The pivot position in a matrix depends on wether row interchanges are used in a row reduction process.
False
T or F. Whenever a system has free variables, the solution set contains many solutions
False
T or F. [-2;5] and [-5;2] span a line in R2
False
T or F. An example of a linear combination of vector v1 and v2 is the vector 1/2v1.
True
T or F. The set Span{u,v} is always visualized as a plane through the origin.
False
T or F. Asking wether the linear system corresponds to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking wether b is in Span{a1 a2 a3}
True
T or F. A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution.
True
The equation Ax=b is consistent if the augmented matrix [A b] has a pivot position in every row
False
If the columns of an mxn matrix A spans Rm, then the equation Ax = b is consistent for each b in Rm
True
If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in Rm, then A cannot have a pivot position in every row.
True
T or F. If the equation Ax=b is inconsistent, then b is not in the set spanned by columns of A.
True
T or F. If the augmented matrix [A b] has a pivot position in every row, then the equation Ax = b is inconsistent.
False
T or F. The equation x=p+tv describes a line through v parallel to p
False
T or F. The homogeneous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable.
False
T or F. If x is a nontrivial solution of Ax=0 then every entry in x is nonzero
False
T or F. The equation Ax=b is homogeneous if the zero vector is a solution.
True
T or F. The solution set of Ax=b is obtained by translating the solution set of Ax=0
False
T or F. The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution
False
T or F. If S is a linearly dependent set, then each vector is a linearly combination of the other vectors in S.
False
T or F. The columns of any 4x5 matrix are linearly dependent
True
T or F. If x and y are linearly independent and if {x,y,z} is linearly dependent, then z is in Spanish {x,y}.
True
T or F. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent
False
T or F. If x and y are linearly independent, and if z is in Spanish {x,y} then {x,y,z} is linearly dependent
True
How many pivot columns must a 7x5 matrix have if its columns are linearly independent? Why?
5
How many pivot columns must a 5x7 matrix have if its columns span R5?
5
T or F. If A is a 3x5 matrix and T is a transformation defined by T(x) = Ax then domain of T is R3
False
T or F. If A is an mxn matrix, then the range of the transformation x-> Axis is Rm.
False
T or F. The codomain of the transformation x-> Ax is the set of all linear combinations of the columns of A
False (range)
T or F. If T: Rn ->Rm is a linear transformation and if c is in Rm, then uniqueness question is “Is c in the range of T?”
False (existence)
T or F. When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.
False
T or F. AB + AC = A(B+C)
True
T or F. At + Bt = (A+B)t
True
T or F. (AB)C = (AC)B
False
T or F. If A and B are nxn and invertible, then A^-1 B^-1 is the inverse of AB
False
T or F. If A is an invertible nxn matrix, then the equation Ax=b is consistent for each b in Rn
True
T or F. If A is invertible then the inverse of A^-1 is A itself
True
T or F. If A can be row reduced to the identity matrix, then A must be invertible
True
T or F. If A is an nxn matrix then the equation Ax=b has at least one solution for each b in R4
False
T or F. If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions
True
T or F. If At is not invertible then A is not invertible
True
T or F. If the columns of A are linearly independent then the columns of A span Rn
True
T or F. If the equation Ax=b has at least 1 solution for each b in Rn then the solution is unique for each b
True
T or F. If there is a b in Rn such that the equation Ax=b is inconsistent then the transformation x-> Ax is not one to one
True
T or F. Can a square matrix with two identical columns be invertible?
No
Is it possible for a 5x5 matrix to be invertible when its columns do not span R5?
No
If A is invertible, can we conclude that the columns of A^-1 are linearly independent?
Yes
If the equation Gx=y has more than one solution for some y in Rn can the columns of G span Rn?
No
T or F. The columns of an invertible nxn matrix form a basis for Rn
True
T or F. Row operations do not affect linear dependence relations among the columns of a matrix
True
T or F. The null spce if an mxn matrix is a subspace of Rn
True
T or F. The column space if matrix A is the set of solutions of Ax=b
False
T or F. If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A
False
T or F. Each line in Rn is a one dimensional subspace of Rn
False
T or F. The dimension of Col A is the number of pivot columns
True
T or F. The dimensions of Col A and Nul A add up to the number of columns of A
True
T or F. If B is a basis for a subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in B
True
T or F. The dimension of the column space of A is Rank A
True
