True / False Flashcards
If a system of linear equations has no free variables, then it has a unique solution.
False
If w is a linear combination of u and v in Rn, then u is a linear combination of v and w.
False
If A is m × n with m pivots, then the linear transformation x → Ax is one to one
False
Let S = {b1 … b2} be linearly independent set in Rn, then S is a basis for Rn
True
If A is a 2x2 matrix such that A^2x = 0, then S is a basis for Rn
False
If A is a symmetric matrix then A is invertible.
False
A system of 4 equations in 5 unknowns can never have a unique solution.
True
The null space of an invertible matrix contains only the zero vector
True
Two subspaces that meet only in the zero vector are orthogonal.
False
If two matrices have the same determinant, they must be similar.
False
If a matrix is diagonalizable, it must have distinct eigenvalues
False
Suppose we apply an elemantary row operation to a matrix A and obtain matrix B. Then A can be obtained by performing an elemantary row operation on B.
True
Elementart row operations on an augmented matrix changes the solution set of the linear system
False
A consistant system of linear equations has on or more solution
True
Suppose a 3 x 5 coefficient matrix for a system has three pivot columns. Is the system consistent?
Yes
Suppose a system of linear equations has a 3 x 5 augmented matrix whose fifth column pivot columns. Is the system consistent?
No
The echelon form of a matrix is unique
False
The vector u resulsts when a vector u - v is added to the vector V
True
An example of a linear combination of vectors v1 & v2 is the vector (1/2)v1
True
Every matrix equation Ax = b corresponds to a vector equation with the same solution set
True
If A is a 3x3 matrix then det (5A) = 5 det (A)
False
if u a v are in R2 and det[uv] = 10, the area of the triangle in the plane with vertices at 0, u, v is 10
False
if A^3 = 0 then det(A) = 0
True
if S is a linearly dependent set, then each vector is a linear combination of the other vectors in S
False
The columns of any 4 x 5 matrix are linearly dependent
True
if v1, v2, v3 and v4 are linearly independent vectors in R4, then {v1, v2, v3} is also linearly independent.
Hint: think about x1v1 + x2v2 + x3v3 + x4v4 = 0
True
The codomain of the transformation x –> Ax is the set of all linear combinations of the columns of A
False
If A is a m x n matrix, then the domain of the transformation x –> Ax is Rn
True
If A is a m x n matrix, then the range of the transformation x –> Ax is Rm
False
If A is a 3x2 matrix, then the transformation x –> Ax with co-domain R3 cannot be onto
True
If A is a 2x3 matrix, then the transformation x –> Ax can’t be one-t-one
True
If A is a 3x2 matrix, then the transformation x –> Ax can’t be one-t-one
False
the canonical vectors of Rn form and orthonormal basis of Rn
True
u, v & w are non zero vectors
If u & v are both orthogonal to w, then u & v are aligned (there exists c such as u =cv)
True
For any nxn matrix A& vectors u, v in Rn we have (Au).(Av) = u.v
False
(1,1,0,1) is a unit vector in the same direction as (3,3,0,3)
False
Not every linearly independent set in Rn is an orthogonal set
True
The first row of AB is the first row of A multiplied on the right by B
True
A square matrix with two identical columns can be invertible
False
If A is a square matrix such that the equation Ax = b is consistent for b in Rn, then A is invertible
True
The matrix {(1,0,0)(0,1,0)(0,a,1)} is invertible regardless the value of a
True
For any matrix A, the products AA^T & A^TA are well defined
True
if A is an nxn matrix, then (A^2)^T = (A^T)^2
True
A vector is an arrow in 3D space
False
A subset H of a vector space V is a subspace of V if the zero vector is in H
False
R2 is a subspace of R3
False
If there is a set {v1,…,v2} that spans V, then dim(V) <= r
True
if dim V = p, then there exists a spanning set of p+1 vectors in V
True
if p >= 2 & dim V = p, then every set of p - 1 non zero vectors is linearly independent
False
