Analysis & Linear Algebra

Vector Spaces and Linear Maps

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1. Bases, dimension, rank–nullity

Read this as a chapter →

Lec-11 Basis for a vector space

NPTEL · Linear Algebra

Lec-12 Dimension of a vector space

NPTEL · Linear Algebra

Lec-16 The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces

NPTEL · Linear Algebra

Rank–nullity — the first move in most linear algebra questions.

Bases, dimension and rank free

In plain words

Dimension counts how many directions you need, and it depends on which scalars you are allowed. Rank–nullity is the accounting: what gets crushed to zero plus what survives equals what you started with.

Rank–nullity

For T : V → W with dim V = n: rank T + nullity T = n.

Consequences: for a square matrix, injective ⇔ surjective ⇔ invertible. In infinite dimensions this fails (the shift operator is injective, not surjective).

Rank facts

rank A = rank Aᵀ = row rank = column rank.
rank is field-independent: a matrix over $\mathbb{Q}$ has the same rank over $\mathbb{R}$ or $\mathbb{C}$ , so nullity computed over either field agrees. (A frequent Part-C item.)
rank(A + B) ≤ rank A + rank B.
rank(AB) ≤ min(rank A, rank B).
Sylvester: rank(AB) ≥ rank A + rank B − n. So two rank-2 matrices in $M_{3}$ can have product of rank 1, but never 0.
rank(A) = rank(AᵀA) over $\mathbb{R}$ .

Dimension formulas

dim(U + W) = dim U + dim W − dim(U ∩ W).
Two subspaces of $\mathbb{R}^{5}$ of dimensions 3 and 3 must intersect in dimension ≥ 1.
dim of the space of n×n matrices: $n^{2}$ ; symmetric n(n+1)/2; skew n(n−1)/2; trace-zero $n^{2} - 1$ ; upper triangular n(n+1)/2.
dim P_n (polynomials of degree ≤ n) = n + 1.

Bases

Every vector space has a basis (needs choice for infinite dimensions). Over a finite field counting matters: the number of bases of 𝔽_q^n is $(q^{n} - 1)(q^{n} - q)\cdots(q^{n} -$ $q^{n-1})$ = |GL_n(𝔽_q)|.

A vector space over any field is never the union of two proper subspaces; over 𝔽 $_{2}$ it can be the union of three.

Key takeaways

Rank–nullity is the first move in almost every question.
Rank does not change when you extend the field.
Sylvester's inequality bounds rank(AB) from below — remember both directions.

Rank–nullity, used the way the exam uses itinteractive

This visual is part of Trap Analysis

Three questions answered from one identity — including the bound most candidates get backwards.

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Exam focus: Rank–nullity plus 'rank is unchanged by field extension' answers most of these. Watch the direction of the rank inequalities for products.

Practice

Part C2025 Juneshow ▾

Let V be the

\mathbb{R}-

vector space of all polynomials with real coefficients. Let

f(x) = x^{2} + x + 1

. Which of the following subsets of V are linearly independent?

A.{f′(x), f(x) − f(x − 1), 1}
B.{f(x + 1) − f(x), f(x) − f(x − 1), 1}
C.{f(x), f′(x), 1}✓ Correct
D.{f(x + 1), f(x − 1), f(x)}✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾

Let 𝔽

_{5}

denote the field with 5 elements. How many 2 × 2 matrices with entries in 𝔽

_{5}

have rank one?

A.125
B.144✓ Correct
C.145
D.480

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

Consider the field 𝔽

_{3}

consisting of 3 elements. Let V be an 𝔽

_{3}-

vector space of dimension 3 and W an 𝔽

_{3}-

vector space of dimension 2. Which of the following statements are true?

A.The number of two dimensional subspaces of V is 13.✓ Correct
B.The number of surjective linear transformations from V to W is 624.✓ Correct
C.The number of one dimensional subspaces of V is 13.✓ Correct
D.The number of linear transformations from V to W is $3^{6}$ .✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

Let V be a real vector space. Suppose that {u,v,w,x,y} ⊆ V is a spanning set of V and that {u,v,x,y} is linearly independent. Which of the following statements are necessarily true?

A.The dimension of V is 4 or 5.✓ Correct
B.If 2u−3v+5w=0, then {v,w,x,y} is a basis of V.✓ Correct
C.If u−6v+7w=0, then the span of {v,w,x} is a 2-dimensional vector space.
D.If u+w=v+x, then {u,v,w,y} is a basis of V.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

Let F_q be a finite field with q elements. For n ≥ 2, let A be a 2n × 2n matrix with entries in F_q such that rank(A) = n. Let W = {v ∈ F_q^(2n) ∣ Av = 0}. Which of the following is necessarily the number of (n+2)-dimensional subspaces of F_q^(2n) that contain W?

A.1
B.(q^(2n) − q^n)(q^(2n) − q^(n+1))/q^n
C.(q^(2n) − 1)(q^(2n) − q)⋯(q^(2n) − q^(n−1))/q^n
D. $(q^n - 1)(q^n - q)/((q^{2} - 1)(q^{2} - q))$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾

Let X be the

\mathbb{R}-

vector space of all twice differentiable real valued functions on [0, 1]. Consider the linear map

\varphi : X \to \mathbb{R}^{3}

defined by

\varphi(f) = (f(1), f'(1), f''(1))

. Which of the following statements is true?

A.The dimension of $X/\ker \varphi$ is 3.✓ Correct
B. $\ker \varphi$ is finite dimensional.
C.The dimension of $X/\ker \varphi$ is 1.
D.X is finite dimensional.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

For a variable x, consider the

\mathbb{Q}-

vector space V = {ax

^{3} +

^{2} +

cx + d |

a, b, c, d \in \mathbb{Q}

}. Further, let A = {

f : V \to \mathbb{Q}

| f is

a \mathbb{Q}-

linear transformation} and B = {f ∈ A | f(1) = 0}. Which of the following statements are true?

A.If f ∈ B, then dim ker f = 3
B.dim B = 3✓ Correct
C.dim A = 4✓ Correct
D.If f ∈ A, then the image of f is a one-dimensional $\mathbb{Q}-$ vector space.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Consider

\mathbb{R}

and

\mathbb{Q}[x]

as vector spaces over

\mathbb{Q}

. Which of the following statements are true?

A.There exists an injective $\mathbb{Q}-$ linear transformation $T : \mathbb{R} \to \mathbb{Q}[x]$
B.There exists an injective $\mathbb{Q}-$ linear transformation $T : \mathbb{Q}[x] \to \mathbb{R}$ ✓ Correct
C.The $\mathbb{Q}-$ vector spaces $\mathbb{Q}[x]$ and $\mathbb{R}$ are isomorphic
D.There do not exist non-zero $\mathbb{Q}-$ linear transformations $T : \mathbb{R} \to \mathbb{Q}[x]$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let V be the subspace spanned by the vectors

v_{1} = (1, 0, 2, 3, 1), v_{2} = (0, 0, 1, 3, 5), v_{3} = (0, 0, 0, 0, 1)

in the real vector space

\mathbb{R}^{5}

. Which of the following vectors are in V?

A.(1, 1, 1, 1, 1)
B.(0, 0, 1, 2, 4)
C.(1, 0, 1, 0, 1)✓ Correct
D.(1, 0, 1, 0, 2)✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let

v = (a, b, c) \in \mathbb{R}^{3}

be a nonzero vector that lies in the orthogonal complement (with respect to the standard inner product) of the row-space of the matrix A = [[2, 2, 7], [3, 1, 4]]. If a, b, c are all integers, then what is the smallest possible value of |a + b + c|?

A.5
B.10✓ Correct
C.15
D.20

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let U denote the span of {eᵗ,

e^{2}

ᵗ,

e^{3}

ᵗ} in the real vector space of continuous functions from

\mathbb{R}

\mathbb{R}

. Consider the

\mathbb{R}-

vector spaces V = {

f : U \to \mathbb{R}

| f is an

\mathbb{R}-

linear transformation} and W = {f ∈ V |

f(e^{3}

ᵗ) = 0}. Which of the following statements is true?

A.Both V and W are infinite-dimensional
B.dim V = 3 and dim W = 1
C.dim V = 3 and dim W = 2✓ Correct
D.V is infinite-dimensional and dim W = 0

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Let

A : \mathbb{R}^{m} \to \mathbb{R}^{n}

be a non-zero linear transformation. Which of the following statements is true?

A.If A is one-to-one but not onto, then m > n
B.If A is onto but not one-to-one, then m < n
C.If A is bijective, then m = n✓ Correct
D.If A is one-to-one, then m = n

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Let A be a 10 × 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true?

A.There exists a vector $v \in \mathbb{R}^{10}$ such that Av ≠ 0 and $A^{2}v = 0$
B.There exists a vector $v \in \mathbb{R}^{10}$ such that $A^{2}v \ne 0$ ✓ Correct
C.A must have a non-zero eigenvalue
D. $A^{7} = 0$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

Let 𝔽 be a finite field and V be a finite dimensional non-zero 𝔽-vector space. Which of the following can NEVER be true?

A.V is the union of 2 proper subspaces.✓ Correct
B.V is the union of 3 proper subspaces.
C.V has a unique basis.
D.V has precisely two bases.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Juneshow ▾

Let B be a 3×5 matrix with entries from

\mathbb{Q}

. Assume that {

v \in \mathbb{R}^{5}

| Bv = 0} is a three-dimensional real vector space. Which of the following statements are true?

A.{ $v \in \mathbb{Q}^{5}$ | Bv = 0} is a three-dimensional vector space over $\mathbb{Q}$ .✓ Correct
B.The linear transformation $T : \mathbb{Q}^{3} \to \mathbb{Q}^{5}$ given by T(v) = Bᵗv is injective.
C.The column span of B is two-dimensional.✓ Correct
D.The linear transformation $T : \mathbb{Q}^{3} \to \mathbb{Q}^{3}$ given by T(v) = BBᵗv is injective.

Solution

Rank does not change under field extension, so rank B = 5 − 3 = 2 over $\mathbb{Q}$ as well: nullity over $\mathbb{Q}$ is 3 and the column span is 2-dimensional.

Bᵗ : $\mathbb{Q}^{3} \to \mathbb{Q}^{5}$ has rank 2 < 3, not injective; BBᵗ has rank 2 < 3, not injective.

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Part Bshow ▾

Let U and W be subspaces of

\mathbb{R}^{7}

with dim U = 4 and dim W = 5. The smallest possible value of dim(U ∩ W) is

A.0
B.1
C.2✓ Correct
D.4

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

T : \mathbb{R}^{5} \to \mathbb{R}^{3}

be a linear transformation. Which of the following must be true?

A.T is not injective.✓ Correct
B.If T is surjective then dim(ker T) = 2.✓ Correct
C.dim(ker T) ≤ 2.
D.T is surjective.

Solution

Rank–nullity gives dim(ker T) + rank(T) = 5, and rank(T) ≤ 3 since the codomain is $\mathbb{R}^{3}$ .

(1) So dim(ker T) ≥ 2 > 0 and T cannot be injective.

(2) Surjective means rank 3, so the kernel has dimension 2.

(3) Fails: the zero map has a 5-dimensional kernel.

(4) Fails for the zero map.

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2. Linear transformations, matrix representation, change of basis

Read this as a chapter →

Lec-14 Linear Transformations

NPTEL · Linear Algebra

Lec19 The Matrix of a Linear Transformation

NPTEL · Linear Algebra

Lec-20 Matrix for the Composition and the Inverse. Similarity Transformations

NPTEL · Linear Algebra

Similarity = same operator in a different basis.

Linear transformations, matrices and similarity free

In plain words

A matrix is a map plus a choice of coordinates. Change the coordinates and the matrix changes while the map does not, so only what stays fixed was ever really about the map.

Change of basis

[T]_B′ = P⁻ $^{1}[T]_B P$ with P the change-of-basis matrix. Two matrices represent the same operator ⇔ they are similar.

Similarity invariants: rank, nullity, trace, determinant, eigenvalues with multiplicities, characteristic polynomial, minimal polynomial, Jordan form.

Not sufficient: equal characteristic and minimal polynomials do not force similarity once n ≥ 4 (nilpotent blocks {2,2} vs {2,1,1}).

Equivalence vs similarity

A ~ B (equivalent, A = PBQ) ⇔ same rank. Every m×n matrix is equivalent to a block [[I_r, 0],[0,0]].
A ≅ B (similar, A = P⁻ $^{1}$ BP) ⇔ same rational/Jordan canonical form.
A ≃ B (congruent, A = PᵀBP) for symmetric matrices ⇔ same signature (Sylvester).

Similar ⇒ equivalent; the converse is far from true.

Invariant subspaces

Over $\mathbb{C}$ every operator has an eigenvector, hence (Schur) is triangularisable: there are invariant subspaces of every dimension 0 ≤ k ≤ n.
Over $\mathbb{R}$ an operator need not have any eigenvector (rotation), but always has an invariant subspace of dimension 1 or 2.

Trace and determinant

trace(AB) = trace(BA), so trace is similarity-invariant; but trace(ABC) ≠ trace(ACB) in general.
AB and BA have the same characteristic polynomial (for square A, B), hence the same eigenvalues — but possibly different minimal polynomials.
det(AB) = det A det B; det(I + AB) = det(I + BA) even for rectangular A, B.

Key takeaways

Similarity invariants: everything intrinsic to the operator; equivalence only sees rank.
Over $\mathbb{C}$ , invariant subspaces of every dimension exist (Schur).
AB and BA: same characteristic polynomial, possibly different minimal polynomial.

Change of basis without sign errorsinteractive

This visual is part of Trap Analysis

Which direction the change-of-basis matrix goes, and the similarity relation that follows.

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Exam focus: Similar matrices are the same operator in different bases: they share rank, trace, determinant, characteristic and minimal polynomials — but sharing those is not enough to be similar.

Practice

Part C2025 Juneshow ▾

Let

M_{2}(\mathbb{R})

denote the space of real 2 × 2 matrices. Let S be the vector subspace of

M_{2}(\mathbb{R})

comprising of all symmetric matrices. Let

F : M_{2}(\mathbb{R}) \to S

be the map defined by F(X) = XXᵀ. Let DF

_A : M_{2}(\mathbb{R}) \to S

be the derivative of F at

A \in M_{2}(\mathbb{R})

. Which of the following statements are true?

A.If AAᵀ = I, then DF $_A : M_{2}(\mathbb{R}) \to S$ is surjective.✓ Correct
B.If AAᵀ = I, then DF $_A : M_{2}(\mathbb{R}) \to S$ need not be surjective.
C.If A is invertible, then DF $_A : M_{2}(\mathbb{R}) \to S$ is surjective.✓ Correct
D.If A is not invertible, then DF $_A : M_{2}(\mathbb{R}) \to S$ is surjective.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

Let V be a real vector space and L(V) denote the space of linear operators on V. Let T∈L(V) be a non-zero operator such that

T^{2}=T

. Consider the subspace W of L(V) spanned by {

I, T^{n} : n

is a positive integer}. Which of the following statements are necessarily true?

A.The set { $S\in{}W : S^{2}=S$ } contains exactly 2 elements.
B. $dim_\mathbb{R}(W) = 2$ .
C.If U∈L(V) is such that $U^{2}=U$ and $(T+U)^{2}=T+U$ , then TU=0.✓ Correct
D.If U∈L(V) is such that $U^{2}=U$ and $(T-U)^{2}=T-U$ , then (TU $)^{2}=$ TU.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let V be a vector space over

\mathbb{R}

. Let

T_{1}, T_{2} : V \to V

be two

\mathbb{R}-

linear transformations such that

T_{1} + T_{2}

and

T_{1} - T_{2}

are linearly independent over

\mathbb{R}

. Consider the following statements: (A) The transformations

T_{1}

and

T_{2}

are linearly independent over

\mathbb{R}. (B)

There exist

\mathbb{R}-

linear transformations

T_{3}, T_{4} : V \to V

such that {

T_{1} + T_{2}, T_{1} - T_{2}, T_{3}, T_{4}

} is linearly independent over

\mathbb{R}

. Which of the following statements is true?

A.(A) is true but (B) is false.
B.(B) is true but (A) is false.
C.Both (A) and (B) are true.✓ Correct
D.Both (A) and (B) are false.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

For a variable x, consider the real vector space V = {ax

^{3} +

^{2} +

cx + d |

a, b, c, d \in \mathbb{R}

}. Let D : V → V be the linear transformation where D(f) is the derivative of f with respect to x, and M : V → V be the linear transformation M(f) = xD(f). Which of the following statements are true?

A.DM ≠ MD.✓ Correct
B.D + M is invertible.
C.DM is invertible.
D.rank(DM) = rank(MD).

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let V (≠ {0}) be a finite dimensional vector space over

\mathbb{R}

and T : V → V be a linear operator. Suppose that the kernel of T equals the image of T. Which of the following statements are necessarily true?

A.The dimension of V is even✓ Correct
B.The trace of T is zero✓ Correct
C.The minimal polynomial of T cannot have two distinct roots✓ Correct
D.The minimal polynomial of T is equal to its characteristic polynomial

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Let V be the real vector space of 2 × 2 matrices with entries in

\mathbb{R}

. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where A = ((2, 0), (0, 1)). What is the characteristic polynomial of T?

A.(x − 2)(x − 1)
B. $x^{2}(x - 2)(x - 1)$
C. $(x - 2)^{2}(x - 1)^{2}$ ✓ Correct
D. $(x^{2} - 2)(x^{2} - 1)$

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

For a variable x, consider the

\mathbb{R}-

vector space V = {

a_{0} + a_{1}x + a_{2}x^{2}

a_{1}, a_{2}, a_{3} \in \mathbb{R}

}. Let T : V → V be the linear transformation defined by T(f) = f + df/dx, where df/dx denotes the derivative of f with respect to x. Which of the following statements is true?

A. $(T^{3} - 3T^{2} + 3T)^2025(x) = x$ ✓ Correct
B. $(T^{3} - 3T^{2} + 3T)^2025(x) = x + 1$
C. $(T^{3} - 3T^{2} + 3T)^2025(x) = 2025! x$
D. $(T^{3} - 3T^{2} + 3T)^2025(x) = 2025! x + 1$

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾

Let

I_{n}

denote the n × n identity matrix. Which one of the following statements is true?

A.If A is a real 3×2 matrix and B a real 2×3 matrix with BA $= I_{2}$ , then AB $= I_{3}$ .
B.Let A = [[3, 3], [1, 2]]. Then there is a matrix B with integer entries such that AB $= I_{2}$ .
C.Let A = [[3, 1], [1, 2]] with entries in $\mathbb{Z}/6\mathbb{Z}$ . Then there is a matrix B with entries in $\mathbb{Z}/6\mathbb{Z}$ such that AB $= I_{2}$ .✓ Correct
D.If A is a real non-zero 3×3 diagonal matrix, then there is a real matrix B such that AB $= I_{3}$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

Let

T : \mathbb{R}^{5} \to \mathbb{R}^{5}

be an

\mathbb{R}-

linear transformation. Suppose that (1, −1, 2, 4, 0), (4, 6, 1, 6, 0) and (5, 5, 3, 9, 0) span the null space of T. Which of the following statements are true?

A.The rank of T is equal to 2.✓ Correct
B.Suppose that for every vector $v \in \mathbb{R}^{5}$ there exists n such that $T^{n}v = 0$ . Then $T^{2}$ must be zero.
C.Suppose that for every vector $v \in \mathbb{R}^{5}$ there exists n such that $T^{n}v = 0$ . Then $T^{3}$ must be zero.✓ Correct
D.(−2, −8, 3, 2, 0) is contained in the null space of T.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾

Which one of the following statements is FALSE?

A.The product of two 2×2 real matrices of rank 2 is of rank 2.
B.The product of two 3×3 real matrices of rank 2 is of rank at most 2.
C.The product of two 3×3 real matrices of rank 2 is of rank at least 2.✓ Correct
D.The product of two 2×2 real matrices of rank 1 can be the zero matrix.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾

Let A be an n × n matrix with complex entries. If n ≥ 4, which one of the following statements is true?

A.A does not have any non-zero invariant subspace in $\mathbb{C}^{n}$ .
B.A has an invariant subspace in $\mathbb{C}^{n}$ of dimension n − 3.✓ Correct
C.All eigenvalues of A are real.
D. $A^{2}$ does not have any invariant subspace in $\mathbb{C}^{n}$ of dimension n − 1.

Solution

Over $\mathbb{C}$ every matrix is triangularisable (Schur), so the span of the first k basis vectors of a triangularising basis is an invariant subspace of every dimension k — in particular n − 3 ≥ 1.

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