# Linear Algebra True/False

False
If x is a nontrivial solution of Ax=0, then every entry in x is nonzero.
Is the statement true or false?
True
The equation Ax = b is homogeneous if the zero vector is a solution.

Is the statement true or false?

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True
The equation x = x2u + x3v with x2 and x3 free (and neither u nor v a multiple of the other), describes a plane through the origin.
False
The solution set of Ax = b is always obtained by translating the solution set of Ax = 0.
False
If A is 3×3 with three pivot positions, then Ax = 0 has a nontrivial solution.
True
If A is 3×3 with three pivot positions, then Ax = b has a solution for every b in R3.
True
If A is 3×3 with two pivot positions, then Ax = 0 has a nontrivial solution.
False
If A is 3×3 with two pivot positions, then Ax = b has a solution for every b in R3.
True
If A is 3×2 with two pivot positions, then Ax = 0 has a nontrivial solution.
False
If A is 3×2 with two pivot positions, then Ax = b has a solution for every b in R3.
True
If A is 2×4 with two pivot positions, then Ax = 0 has a nontrivial solution.
True
If A is 2×4 with two pivot positions, then Ax = b has a solution for every b in R2.
True
Two vectors in R3 are linearly dependent if and only if they both lie on same line through the origin.
True
If a matrix has more rows than columns, then its rows must be linearly dependent.
True
If vectors x, y, and z are such that z belongs to span{x,y}, then the collection {x,y,z} must be linearly dependent.
False
If the columns of a matrix are linearly dependent, then the matrix must have more columns than rows.
True
If a collection of vectors in Rn contains the zero vector, then the collection must be linearly dependent.
False
If the third column of a 4×4 matrix is not a linear combination of the other three columns, then the columns of this matrix must be linearly independent.
True
If the set of vectors {v1, v2, v3, v4, v5} is linearly independent, then so is {v1, v2, v4, v5}.
True
Elementary row operations on an augmented matrix never change the
solution set of the associated linear system
False
Two matrices are row equivalent if they have the same number of rows.
False
Let A be an mxn matrix, and assume m > n. Additionally, assume that the
matrix A has n pivots. Then for any vector b ∈ Rm, the equation Ax = b is consistent.
True.
If for some matrix A, and some vectors x,b, we have Ax = b, then b is a
linear combination of the column vectors of A.
True
If A and B are both invertible nxn matrices, then AB is invertible.
True
Let A and B be nxn matrices. Assume that AB = In. Then, BA = In.
The homogeneous equation Ax = 0 is always consistent. this is
because after row reducing you will never have a row of all zeros and a nonzero in the
augmented part, (because the augmented part is all zeros). In fact the zero vector is always
a solution to a homogeneous equation.
Let A be an mxn matrix. Then, the homogeneous equation Ax = 0 is consistent
if and only if the augmented matrix [ A | 0 ] has a pivot in every row.
True.
Let A be an mxn matrix and let A^T be its transpose. Then A^T A is an nxn
matrix so its determinant is well defined. In fact, the determinant det(A^T A) is greater than
or equal to zero.
False
The equation Ax = b has at least one solution
for each b ∈ R
n.
True
If the equation Ax = 0 has nontrivial solutions
then A has fewer than n pivot positions.
True
If AT
is not invertible then A is not invertible
True
If there is an n × n matrix D such that
AD = I then there is also an n × n matrix
C such that CA = I.
True
If the columns of A are linearly independent
then they span Rn
True
If the equation Ax = b has at least one
solution for each b ∈ R
n, then the solution
is unique for each b.
True
If the linear transformation x → Ax maps
Rn onto Rn, then A has n pivot positions
True
If there is b ∈ R
n such that the equation
Ax = b is inconsistent, then the transformation
x→ Ax is not one-to-one
True
Row operations do not affect linear dependence
relations among the columns of
a matrix. 1
False
A product of invertible n×n matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
True
If a matrix A is invertible, then the inverse of A^−1 is A itself.
False
If A = [a b; c d] and ad=bc, then A is invertible.
True
If a matrix A can be row reduced to the identity matrix, then A must be invertible.
False
If a matrix A is invertible, then the same elementary row operations that reduce A to the identity I also reduce A^−1 to I.
True
If there is an n×n matrix D such that AD=I, then DA=I.

Note: The matrix A is n×n.

True
If the columns of an n×n matrix A are linearly independent, then the columns of A span Rn
True
If the equation Ax=b has at least one solution for each b in Rn, then the solution is unique for each b.

Note: The matrix A is n×n.

False
If the linear transformation x ->Ax maps Rn to Rn, then A always has n pivot positions.

Note: The matrix A is n×n.

True
If there is a b in Rn such that the equation Ax=b is inconsistent (i.e. has no solution), then the transformation x->Ax is not one-to-one.

Note: The matrix A is n×n.

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