Linear Final True or False

Linear system with four equations and seven variables must have infinitely many solutions

False. A system can always be inconsistent.

The plane 2x + y - z = 3 contains the point (1,1,1)

False. Verify by plugging in x = 1, y = 1, z = 1.

The set of all solutions (x, y, z) to the equation x - y - z = 0 is a line in R3

False. It would have one pivot and two free variable columns, resulting in a plane.

The equation 3x + ln(2)y = π is a linear equation in x and y

True. ln(2) and π are constants, fitting the ax + by = c format.

Possible for a single linear equation with 2 unknowns to have exactly one solution

False. There must be a free variable column, ensuring no unique solution.

A system of six equations with eight unknowns has the largest possible number of pivots equal to 8

False. This 6x9 matrix has at most 6 pivots.

A 'tall matrix' can never have a pivot in every row

True. You can only have one pivot per column.

Two non-collinear vectors always span a plane

True.

If the bottom row of an augmented matrix is (0, 1, 3, 1), then the system has no solution

False. There are infinitely many solutions since the third column does not contain a pivot.

The following matrices are in reduced row echelon form (RREF)

False. Pivot columns should not contain entries besides the pivot.

If the solution to a system of equations is (4 - 2z, -3 + z, z), then (4, -3, 0) is a solution

True. Check by plugging in the values.

If the bottom row of a matrix in reduced row echelon form contains all 0s to the left and a nonzero to the right, then the system has no solution

True. Corresponds to the equation 0 = nonzero.

Span{a1, a2} contains only the line through a1 and the origin, and a2 and the origin

False. Span{a1, a2} contains all linear combinations, not just the lines.

The solution set of the matrix equation [a1 a2 a3 | b] matches the equation x1a1 + x2a2 + x3a3 = b

True. Both represent the same linear systems.

There are exactly three vectors in the set {a1, a2, a3}

True. It's a set, not a span.

Asking if the equation [a1 a2 a3 | b] has a solution means checking if b is in Span{a1, a2, a3}

True. Both relate to the same linear equations.

There are exactly three vectors in Span{a1, a2, a3}

False. There are infinitely many, consisting of all multiples.

A = . There is not a solution for every b in R3 given Ax = b

True. Less dimensions in A leads to no guarantee of solutions.

The solution set of the augmented matrix [a1 a2 a3 | b] is the same as Ax = b, if A = [a1 a2 a3]

True.

If the equation Ax = b is inconsistent, then b is not in the span of the columns of A

True.

The equation Ax = b is a vector equation

False. It is specifically called a matrix equation.

Every matrix equation Ax = b corresponds to a vector equation with the same solution set

True.

If A is an mxn matrix and Ax = b is inconsistent for some b in Rm, A cannot have a pivot in every row

True.

The equation Ax = b is consistent if [A | b] has a pivot in every row

False. A pivot in the last column could lead to inconsistency.

The equation Ax = b is homogeneous if the zero vector is a solution

True.

The solution set of a consistent inhomogeneous system Ax = b is derived by translating the solution set of Ax = 0

True.

There is a vector such that the solutions set = is the z-axis

False.

The homogeneous system Ax = 0 has a trivial solution if and only if there is at least one free variable

False. The trivial solution is always there.

A homogeneous linear system is always consistent

True.

If x is a nontrivial solution of Ax = 0, every entry of x is nonzero

False. Only one entry needs to be nonzero.

Let A be a matrix with more rows than columns. Columns of A must be linearly dependent

False. Columns may be independent.

Let A be a matrix with linearly independent columns. Then Ax = b has a solution for all b if it is square

True.

The columns of a matrix with m < n must be linearly dependent

True.

The columns of matrix A are linearly independent if Ax = b has the trivial solution

False.

If S is a set of linearly dependent vectors, every vector in S can be expressed as a linear combination of others

False. Only one needs to be expressible.

Two vectors are linearly dependent if and only if they are collinear

True.

If a set S of vectors contains fewer vectors than entries, the set must be linearly independent

False.

Let V be the subset of R3 with abc = 0

True. Zero vector is always included.

V is closed under addition, meaning if u and v in V then u + v is in V

False.

V is closed under scalar multiplication, meaning if u is in V, then cu is also in V

True.

V is a subspace of R3

False.

The solution set of a system of m homogeneous equations in n unknowns is a subspace of Rn

True.

The column space of an m x n matrix is a subspace of Rm

True.

If B is an echelon form of matrix A, the pivot columns of B form a basis for the column space of A

False.

The null space of an m x n matrix is a subspace of Rm

False.

Any set of n linearly independent vectors in Rn is a basis for Rn

True.

Let A be an 8 x 9 matrix. What must m and n be if T(x) = Ax?

m = 9 and n = 8.

Let T be a one-to-one matrix transformation from Rn to Rm. Then n < m

False.

Let T : R2 → R2 be the function given by T(cv) = cT(v) for all v

False.

T(u + v) = T(u) + T(v) for all u and v

False.

T is a linear transformation

False.

The transformation T defined by T(x1, x2, x3) = (x1, x2, -x3) is a linear transformation

True.

The transformation T defined by T(x1, x2) = (4x1 - 2x2, 3|x2|) is a linear transformation

False.

The transformation T defined by T(x1, x2) = (2x1 - 3x2, x1 + 4, 5x2) is a linear transformation

False.

The transformation T defined by T(x1, x2, x3) = (x1, 0, x3) is a linear transformation

True.

The transformation T defined by T(x1, x2, x3) = (1, x2, x3) is a linear transformation

False.

For any matrix A, there exists a matrix B such that A + B = 0

True.

If A is a 5x4 matrix and B is a 4x3 matrix, the entry of AB in 3rd row / 2nd column is found by multiplying the 3rd column of A by the 2nd row of B

False.

For any matrix A, we have 2A + 3A = 5A

True.

For any matrices A and B, if the product AB is defined, then BA is also defined

False.

If A is an mxn matrix and B is an nxm matrix, then both AB and BA are defined

True.

Suppose A and B are invertible nxn matrices. (A + B)^2 = A^2 + B^2 + 2AB

False.

A^7 is invertible

True.

A + B is invertible

False.

(AB)^-1 = A^-1B^-1

False.

(In - A)(In + A) = In - A^2

True.

If the linear transformation T(x) = Ax is onto, it is also one-to-one

True.

If the linear transformation T(x) = Ax is one-to-one, then columns of A form a linearly dependent set

False.

If Ax = 0 has a nontrivial solution, A has fewer than n pivots

True.

If -A is not invertible, then A is also not invertible

True.

If A is invertible, then Ax = b has a unique solution for all b in Rn

True.

If A^2 is row equivalent to the nxn identity matrix, the columns of A span Rn

True.

A square matrix with duplicated columns can be invertible

False.

The product of any two invertible matrices is invertible

True.

The following transformations from R3 to R3 are invertible: Projection onto the y-axis

False.

Identity transformation: T(v) = v for all v

True.

Projection onto the xz-plane is invertible

False.

Rotation about the z-axis by π is an invertible transformation

True.

Dilation by a factor of 8 is an invertible transformation

True.

Reflection in the origin is an invertible transformation

True.

If A is a 3x3 matrix and λ has the property Ax = λx for some nonzero x

A - λI is not invertible.

A - λ is invertible

False.

A is not invertible

False.

The absolute value of the determinant of A equals the volume of the parallelepiped determined by the columns of A

True.

The determinant of a triangular matrix is the sum of the main diagonal's entries

False.

A determinant of an nxn matrix can be defined as a sum of multiples of determinants of (n-1)x(n-1) submatrices

True.

The i,j minor of a matrix A is the matrix Aij obtained by deleting row i and column j from A

True.

The cofactor expansion of det A along the first row equals the expansion along any other row

True.

If the columns of A are linearly independent, then det A = 0

False.

A row replacement operation does not affect the determinant of a matrix

True.

If det A is zero, then two columns of A must be the same, or all elements in a row/column of A are zero

False.

If two columns of A are the same, the determinant is zero

True.

det(A + B) = det(A) + det(B)

False.

If a matrix has a determinant of zero, it is invertible

False.

Row reduction does not change whether the determinant is zero

True.

The determinant of a transposed matrix is the same as the original matrix

True.

Applying a R2 to R2 transformation with a matrix determinant of 2 will double the area of any transformed shape

True.

Det(AB) = Det(A)・Det(B)

False, if A and B are not square matrices.

Eigenvectors with different corresponding eigenvalues are always linearly independent

True.

An nxn matrix can have more than n eigenvalues

False.