Math 121A T/F questions with 100% accurate solutions + rationales 2026

Every vector space contains a zero vector.

True; VS 3 ( ∃ 0 ∈ V s. t. x + 0 = x, ∀ x ∈ V)

A vector space may have more than one zero vector.

False; VS 3 ( ∃ 0 ∈ V s. t. x + 0 = x, ∀ x ∈ V)

In any vector space, ax = bx implies that a = b.

False; x = 0

In any vector space, ax = ay implies that x = y.

False; a = 0

A vector in Fⁿ may be regarded as a matrix in Mₙx₁ (F).

True; a single row matrix may be regarded as a row vector

An mxn matrix has m columns and n rows.

False; m rows and n columns

In P(F), only polynomials of the same degree may be added.

False; (x⁵ + 1) + (x) = x⁵ + x + 1

If f and g are polynomials of degree n, then f + g is a polynomial of degree n.

False; f = x² and g = -x²; f + g = x² + (-x²) = 0

If f is a polynomial of degree n and c is a nonzero scalar, then cf is a polynomial of degree n.

True; x * c = cx

A nonzero scalar of F may be considered to be a polynomial in P(F) having degree zero.

True; F = scalar

Two functions in F(S, F) are equal if and only if they have the same value at each element of S.

True; F(x) = S(x) ∀x

If V is a vector space and W is a subset of V that is a vector space, then W is a subspace of V.

True; since vector addition and scalar multiplication will also hold for W ⊆ V

The empty set is a subspace of every vector space.

False; every subspace must contain zero vector

If V is a vector space other than the zero vector space, then V contains a subspace W such that W ≠ V.

True; W would be a proper subspace

The intersection of any two subsets of V is a subspace of V.

False; A = {1, 2} and B = {2, 3}, A∩B = {2} does not include zero

An n x n diagonal matrix can never have more than n nonzero entries.

True; n diagonal positions

The trace of a square matrix is the product of its diagonal entries.

False; trace is the sum of diagonal entries

Let W be the xy-plane in R³; that is, W = {(a₁, a₂,0): a₁, a₂ ∈ R}. Then W = R²

False; Different dimensions

The zero vector is a linear combination of any nonempty set of vectors.

True; 0 * v = 0

The span of {∅} is {∅}

False; span(∅) = {0}

If S is a subset of vector space V, then span(S) equals the intersection of all subspaces of V that contain S.

True; span is smallest subspace containing S (Thrm)

In solving a system of linear equations, it is permissible to multiply an equation by any constant.

False; constant must be nonzero

In solving a system of linear equations, it is permissible to add any multiple of one equation to another.

True; elementary row operation

Every system of linear equations has a solution

False; f = 2f

If S is a linearly dependent set, then each vector in S is a linear combination of the other vectors in S.

False; { a, b, 2a} (a ≠ b) only a and 2a are related

Any set containing the zero vector is linearly dependent

True; can multiply any vector by 0 to get zero vector

The empty set is linearly dependent

False; empty set is linearly independent

Subsets of linearly dependent sets are linearly dependent

False; {a, b, 2a} has subset {a, b}

Subsets of linearly independent sets are linearly independent

True; still no nontrivial linear combination can be formed

If a₁x₁+a₂x2₂+⋯+aₙxₙ=0 and x₁, . . . , xₙ are linearly independent, then all scalars a₁, . . . , aₙ = 0

True; definition of linear independence

The zero vector has no basis

False; ∅ is basis for {0}

Every vector space that is generated by a finite set has a basis

True; finite sets can be reduced to a basis

Every vector space has a finite basis

False

A vector space cannot have more than one basis

False

If a vector space has a finite basis, then the number of vectors in every basis is the same.

True

The dimension of Pₙ(F) is n

False

The dimension of mxn matrix is m + n

False

Suppose V is finite-dimensional, S₁ is linearly independent, and S₂ spans V. Then S₁ cannot contain more vectors than S₂.

True

If S generates V, then every vector in V can be written as a linear combination of vectors in S in only one way.

False

Every subspace of a finite-dimensional space is finite-dimensional.

True

If V has dimension n, then V has exactly one subspace with dimension 0 and exactly one subspace with dimension n.

True

If V has dimension n, and S⊆V has n vectors, then S is linearly independent iff S spans V

True

Any linear operator on an nnn-dimensional vector space that has fewer than nnn distinct eigenvalues is not diagonalizable.

False

Two distinct eigenvectors corresponding to the same eigenvalue are always linearly dependent.

False

If λ is an eigenvalue of T, then each vector in Eλ is an eigenvector of T

False

If λ₁ and λ₂ are distinct eigenvalues of a linear operator T, then Eλ₁ ∩ Eλ₂ = {0}

True

If B={v₁, ... ,vₙ} is a basis of eigenvectors of A, and Q has these vectors as columns, then Q⁻¹AQ is diagonal

True

T is diagonalizable if and only if the multiplicity of each eigenvalue equals dim(Eλ)

True

Every diagonalizable linear operator on a nonzero vector space has at one eigenvalue.

True

If V = W₁⊕W₂⊕. . . ⊕Wₙ, then Wi ∩ Wj = {0} for i ≠ j

True

Every linear operator on an n-dimensional vector space has n distinct eigenvalues.

False

If a real matric has one eigenvector, then it has an infinite number of eigenvectors.

True

There exists a square matrix with no eigenvectors

True

Eigenvalues must be nonzero scalars.

False

Any two eigenvectors are linearly independent

False

The sum of two eigenvalues of T is also an eigenvalue of T

False

Linear operators on infinite-dimensional vector spaces never have eigenvalues.

False

An nxn matrix A is similar to a diagonal matrix if and only if there is a basis of Fⁿ consisting of eigenvectors of A.

True

Similar matrices always have the same eigenvalues

True

Similar matrices always the same eigenvectors

False

The sum of two eigenvectors of T is always an eigenvector of T

False

If E is elementary, then det(E) = ±1

False

For any A, B ∈ Mₙ(F), det(AB) = det(A)det(B)

True

A nxn matrix M is invertible if and only if det(M) = 0

False

An nxn matrix M has rank n if and only if det(M) ≠ 0

True

For any nxn matrix A, det(A) = -det(A)

False

Determinant can be computed by cofactor expansion along any column.

True

The function det: M₂(F) → F is a linear transformation

False

The determinant of a square matrix can be evaluated by cofactor expansion along any row

True

If two rows of a square matrix A are identical, then det(A) = 0

True

If B is obtained from A by interchanging any two rows, then det(B) = -det(A)

True

If B is obtained from A by multiplying a row of A by a scalar, then det(B) = det(A).

False

If B is obtained from A by adding k times row i to row j, then det(B) = kdet(A)

False

If A ∈ Mₙ(F) has rank n, then det(A) = 0

False

The determinant of an upper triangular matrix equals the product of its diagonal entries

True

The determinant of a 2x2 matrix is a linear function of each row of the matrix when the other row is fixed

True

If A∈M₂(F) and det(A) = 0, then A is invertible

False

If u and v are vectors in R² emanating from the origin, then the area of the parallelogram having u and v as adjacent sides is det(u, v)

False

A coordinate system is right handed if and only if its orientation equals 1

True

The rank of a matrix is equal to the number of its nonzero columns

False

The product of two matrices always has rank equal to the lesser of the ranks of the two matrices

False

The mxn zero matrix is the only matrix having rank 0

True

Elementary row operations preserve rank

True

Elementary column operations do not necessarily preserve rank

False

The rank of a matrix is equal to the maximum number of linearly independent rows in the matrix

True

The inverse of a matrix can be computed exclusively by means of elementary row operations

True

The rank of an nxn matrix is at most n

True

An nxn matrix having rank n is invertible

True

An elementary matrix is always square

True

The only entries of an elementary matrix are zeros and ones

False

The nxn identity matrix is an elementary matrix

False

The product of two nxn elementary matrices is an elementary matrix

False

The inverse of an elementary matrix is an elementary matrix

True

The sum of two nxn elementary matrices is an elementary matrix

False

The transpose of an elementary matrix is an elementary matrix

True

If B is a matrix that can be obtained by performing an elementary row operation on a matrix A, then B can also be obtained by performing an elementary column operation on A.

False

If B is a matrix that can be obtained by performing an elementary row operation on a matrix A, then A can be obtained by performing an elementary row operation on B.

True

If T is linear, then T preserves sums and scalar products

True

If T (x + y) = T (x) + T (y). then T is linear.

False

T is 1-1 if and only if the only vector x such that T(x) = 0 is x = 0

True