Matrices True/False
Every matrix is row equivalent to a unique matrix in echelon form.
False
If A and B are 3 × 3 matrices and B = [b1 b2 b3] , then the product AB is given by AB = [Ab1 + Ab2 + Ab3]
False
If A is an m × n matrix and the equation Ax = b is consistent for some b in R^m, then the columns of A span R^m.
False
If a system of linear equations has no free variables, then it has a unique solution.
False
The columns of a matrix A are linearly independent when the equation Ax = 0 has the solution x = 0.
False
The equality (ABC) T = C T A T B T holds for all n × n matrices A, B, and C.
False
The homogeneous equation Ax = 0 has the trivial solution x = 0 if and only if the equation has at least one free variable.
False
The matrices A = a 1 0 1 B = 1 1 0 b have the property AB = BA , for all values of a and b.
False
The sum of the vector u − v and the vector v is the vector v.
False
When u = −2 5 v = −5 2 then the vectors in Span{u, v} lie on a line through the origin.
False
When u, v are nonzero vectors, then Span{u, v} contains only the line through u and the origin, and the line through v and the origin
False
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
True
A consistent system of linear equations has one or more solutions.
True
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
An example of a linear combination of vectors v1 and v2 is the vector −2v1.
True
Any list of five real numbers is a vector in R^5
True
For every matrix equation Ax = b there corresponds a vector equation having the same solution set.
True
If A is an n × n matrix, then (A^2)^T = (A^T)^2
True
If u and v are linearly independent and w is in Span{u, v}, then the set {u, v, w} is linearly dependent.
True
The equation Ax = b is homogeneous if the zero vector is a solution.
True
The four points (0, 0, 0), (4, −2, 2), (−3, 1, −3), (−2, −3, −13), in 3-space are co-planar, i.e., lie on a plane.
True
The matrix equation A x = 1 1 −3 −3 −2 11 1 2 0 x = −1 −2 1
True
The points (2, −1), (3, −2), (0, 1) in the plane are collinear, i.e., lie on a straight line.
True
The product of the m × n matrix A = [a1 a2 . . . an] and the vector x = x1 x2 xn in R^n is the vector x1a1 + x2a2 + . . . + xnan in R^m
True
There are 2 × 2 matrices A, B, and C for which AB = AC and A =/ 0, but B =/ C.
True
