7. Linear Algebra

Vector space with examples

A vector space is an algebraic structure formed by a set of vectors combined with two operations satisfying ten specific axioms of addition and scalar multiplication.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

📖 Topic progress

○ Mark as complete

Definition and statement

Definition of a Vector Space over a Field $\mathbb{F}$

Let $V$ be a non-empty set and $\mathbb{F}$ be a field (typically $\mathbb{R}$ or $\mathbb{C}$). $V$ is called a vector space over $\mathbb{F}$ if there exist two compositions: Vector Addition ($+: V \times V \to V$) and Scalar Multiplication $(\cdot: \mathbb{F} \times V \to V)$ such that for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $\alpha, \beta \in \mathbb{F}$: (1) $\mathbf{u} + \mathbf{v} \in V$, (2) $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$, (3) $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$, (4) $\exists \mathbf{0} \in V$ s.t. $\mathbf{u} + \mathbf{0} = \mathbf{u}$, (5) $\exists -\mathbf{u} \in V$ s.t. $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$, (6) $\alpha \mathbf{u} \in V$, (7) $\alpha(\mathbf{u} + \mathbf{v}) = \alpha \mathbf{u} + \alpha \mathbf{v}$, (8) $(\alpha + \beta)\mathbf{u} = \alpha \mathbf{u} + \beta \mathbf{u}$, (9) $(\alpha \beta)\mathbf{u} = \alpha (\beta \mathbf{u})$, (10) $1 \cdot \mathbf{u} = \mathbf{u}$.

📐 Geometric Interpretation

Vectors represent directed line segments or coordinate tuples in $\mathbb{R}^n$ that obey parallelogram laws of addition and uniform scaling without altering the origin.

⚙️ Algebraic & Structural Meaning

It provides the foundational algebraic framework for linear transformations, eigenvalues, inner product spaces, and solving systems of linear equations.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Closure under Vector Addition	`Set $V$ and operation $+$`	Strictly Mandatory / Necessary	If the sum of two elements leaves the subset (e.g., vectors with non-negative components), the structure fails to be a vector space.
Closure under Scalar Multiplication	`Field $\mathbb{F}$ and operation $\cdot$`	Strictly Mandatory	Scaling an element by a scalar in $\mathbb{F}$ must yield an element inside $V$; if it produces elements outside, subspace status is lost.
Existence of Zero Vector and Inverses	`Internal to $V$`	Essential for group structure under addition	Without the zero vector or additive inverses, cancellation laws fail and linear independence cannot be properly defined.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Subspace Test for Homogeneous Equations

$$W = \{ (x, y, z) \in \mathbb{R}^3 \mid ax + by + cz = 0 \}$$

Rule: Any homogeneous linear equation in $n$ variables defines a subspace of dimension $n-1$. Non-homogeneous equations ($= k$ where $k \neq 0$) never form subspaces because they fail to contain the zero vector.

Polynomial Degree Restriction

$$W = \{ p(x) \in P_n \mid deg(p) = k \text{ or } p(0) = 1 \}$$

Rule: Sets of polynomials defined by a fixed exact degree $deg(p) = k$ or non-zero constant evaluation (e.g., $p(0) = 1$) are NOT subspaces. Only linear homogeneity conditions (like $p(1) = 0$) form subspaces.

Matrix Subspace Dimensions

$$M_{n \times n}(\mathbb{R}), \quad \text{Sym}_n(\mathbb{R}), \quad \text{Skew}_n(\mathbb{R})$$

Rule: Dimension of symmetric matrices is $\frac{n(n+1)}{2}$, skew-symmetric matrices is $\frac{n(n-1)}{2}$, and diagonal matrices is $n$.

Application: Direct numerical questions in LT Grade and UP TGT mathematics.

Intersection of Subspaces

$$W_1 \cap W_2 \text{ is always a subspace}$$

Rule: The intersection of any arbitrary family of subspaces is always a subspace. However, the union $W_1 \cup W_2$ is a subspace if and only if $W_1 \subseteq W_2$ or $W_2 \subseteq W_1$.

Function Space Subspaces

$$W = \{ f: \mathbb{R} \to \mathbb{R} \mid f(-x) = f(x) \text{ (even functions)} \}$$

Rule: The set of all even functions on $\mathbb{R}$ forms a subspace of $F(\mathbb{R}, \mathbb{R})$, whereas the set of all odd functions also forms a subspace. Their direct sum yields the full space of functions.

Application: Advanced PGT question pattern.

Rapid Zero-Vector Elimination

$$0 \cdot \mathbf{u} = \mathbf{0}$$

Rule: Always check if the zero vector (zero tuple, zero polynomial, zero matrix) belongs to the given subset. If $\mathbf{0} \notin W$, instantly eliminate the option.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

The Non-Homogeneous Constant Trap

Common Pitfall: Candidates assume subsets defined by equations with non-zero constants (e.g., $x + y + z = 5$) are subspaces.

Counterexample: $W = \{(x, y) \in \mathbb{R}^2 \mid x + y = 1\}$

Why it fails: The zero vector $(0,0)$ does not satisfy $0 + 0 = 1$, violating the mandatory zero-vector axiom.

🛡️ Exam Defense: Check the constant term first. If it is non-zero in a linear equation, it is never a subspace.

⚠️ COMMON ERROR #02

Union of Subspaces Fallacy

Common Pitfall: Assuming the union of two subspaces is automatically a subspace.

Counterexample: Let $W_1 = \{(x, 0) \mid x \in \mathbb{R}\}$ and $W_2 = \{(0, y) \mid y \in \mathbb{R}\}$ in $\mathbb{R}^2$.

Why it fails: (1,0) \in W_1$ and $(0,1) \in W_2$, but their sum $(1,1)$ does not lie in $W_1 \cup W_2$.

🛡️ Exam Defense: Union of subspaces is a subspace $\iff$ one is contained within the other.

Comparison

Concept / Structure	Underlying Set / Field	Key Axiom / Condition	Dimension Formula (Standard)	Exam Identification Clue
Euclidean Space $\mathbb{R}^n$	Tuples of real numbers over $\mathbb{R}$	Standard coordinatewise operations	$n$	Standard Cartesian coordinates
Polynomial Space $P_n$	Polynomials of degree $\le n$	Standard polynomial addition and scalar multiplication	$n+1$	Degree constraint $\le n$
Matrix Space $M_{m \times n}$	Matrices of size $m \times n$ over $\mathbb{R}$	Matrix addition and scalar multiplication	$m \times n$	Rectangular arrays of numbers
Symmetric Matrices $\text{Sym}_n$	Square matrices where $A^T = A$	Closed under addition and scalar multiplication	\frac{n(n+1)}{2}	Condition $a_{ij} = a_{ji}

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Which of the following subsets of $\mathbb{R}^3$ is a subspace of $\mathbb{R}^3$?

(A) $\{(x, y, z) \mid x + y + z = 1\}$(B) $\{(x, y, z) \mid xyz = 0\}$(C) $\{(x, y, z) \mid x - 2y + 3z = 0\}$(D) $\{(x, y, z) \mid x \ge 0\}$

📚 Method A: Formal Textbook Derivation

Step 1: Check zero vector. Only (C) contains $(0,0,0)$ since $0 - 2(0) + 3(0) = 0$. Step 2: Verify closure under addition and scalar multiplication for homogeneous linear equations. Thus, (C) is a subspace.

⚡ Method B: 25-Second Exam Shortcut

Instantly discard (A) due to non-zero constant 1, discard (B) due to non-linearity ($xyz$), discard (D) due to inequality ($x \ge 0$). Select (C) in 5 seconds.

💡 Tip: Homogeneous linear equations always form subspaces.

CANONICAL PROBLEM 02 Dual-Method Breakdown

What is the dimension of the vector space of all symmetric matrices of order $n$ over $\mathbb{R}$?

(A) $n^2$(B) $\frac{n(n+1)}{2}$(C) $\frac{n(n-1)}{2}$(D) $n^2 - n$

📚 Method A: Formal Textbook Derivation

A symmetric matrix $A = [a_{ij}]$ satisfies $a_{ij} = a_{ji}$. The main diagonal has $n$ independent entries. The upper triangle has $\frac{n(n-1)}{2}$ independent entries. Total = $n + \frac{n(n-1)}{2} = \frac{n(n+1)}{2}$.

⚡ Method B: 25-Second Exam Shortcut

Recall standard dimension formulas: Sym = $n(n+1)/2$, Skew-sym = $n(n-1)/2$, Full matrix = $n^2$.

💡 Tip: Direct memory-based question frequently asked in UP PGT.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: Which of the following is a vector space over $\mathbb{R}$?

A The set of all invertible $n \times n$ matrices

B The set of all polynomials of degree exactly $n$

C The set of all continuous functions $f: [0, 1] \to \mathbb{R}$

D The set of all $(x, y) \in \mathbb{R}^2$ such that $xy \ge 0$

Derivation & Analysis:

Continuous functions are closed under addition and scalar multiplication and contain the zero function, forming a valid vector space.

MCQ 2: The dimension of the vector space of all diagonal matrices of order $n$ is:

A $n^2$

B $n$

C $n(n-1)/2$

D $n(n+1)/2$

Derivation & Analysis:

A diagonal matrix has entries only on the main diagonal, giving $n$ independent free variables.

MCQ 3: Let $W_1$ and $W_2$ be subspaces of a vector space $V$. Then $W_1 \cup W_2$ is a subspace of $V$ if and only if:

A $W_1 \cap W_2 = \{0\}$

B $W_1 \subseteq W_2$ or $W_2 \subseteq W_1$

C $W_1 + W_2 = V$

D $W_1$ and $W_2$ are disjoint

Derivation & Analysis:

The union of two subspaces is a subspace if and only if one is contained in the other.

MCQ 4: What is the dimension of the vector space $P_n$ of polynomials of degree at most $n$?

A $n$

B $n-1$

C $n+1$

D $2n$

Derivation & Analysis:

The standard basis is $\{1, x, x^2, \dots, x^n\}$, which contains $n+1$ elements.

MCQ 5: Which subset of $\mathbb{R}^3$ is NOT a subspace?

A $\{(x, y, z) \mid x + 2y - 3z = 0\}$

B $\{(x, y, z) \mid x = y = z\}$

C $\{(x, y, z) \mid z = 0\}$

D $\{(x, y, z) \mid x + y + z = 1\}$

Derivation & Analysis:

The subset defined by $x+y+z=1$ does not contain the zero vector $(0,0,0)$. Thus, it is not a subspace.

MCQ 6: The dimension of the vector space of all skew-symmetric matrices of order $n$ is:

A $\frac{n(n+1)}{2}$

B $\frac{n(n-1)}{2}$

C $n^2 - n$

D $n^2$

Derivation & Analysis:

For a skew-symmetric matrix, diagonal entries are zero and $a_{ij} = -a_{ji}$, leaving $\frac{n(n-1)}{2}$ independent entries.

MCQ 7: If $V$ and $W$ are subspaces of dimensions 4 and 5 respectively in a 8-dimensional vector space, what is the minimum possible dimension of $V \cap W$?

A 0

B 1

C 2

D 3

Derivation & Analysis:

Using dimension formula $\dim(V+W) = \dim(V) + \dim(W) - \dim(V \cap W)$, since $\dim(V+W) \le 8$, we get $4 + 5 - \dim(V \cap W) \le 8 \implies \dim(V \cap W) \ge 1$.

MCQ 8: The set of all solutions to a homogeneous system of $m$ linear equations in $n$ variables over $\mathbb{R}$ forms:

A A field

B A subspace of $\mathbb{R}^n$

C A basis of $\mathbb{R}^n$

D An empty set

Derivation & Analysis:

The solution space of a homogeneous system of linear equations always forms a subspace of $\mathbb{R}^n$, known as the null space.

MCQ 9: Which of the following mappings defines a valid vector space over $\mathbb{R}$ under standard operations?

A The set of all complex numbers $\mathbb{C}$ considered over field $\mathbb{R}$

B The set of all positive real numbers

C The set of polynomials with integer coefficients

D The set of step functions

Derivation & Analysis:

Complex numbers form a 2-dimensional vector space over the field of real numbers $\mathbb{R}$.

Q10

MCQ 10: If $W_1$ and $W_2$ are two subspaces of $\mathbb{R}^5$ with dimensions 3 and 4 respectively, what is the maximum possible dimension of $W_1 \cap W_2$?

A 2

B 3

C 4

D 5

Derivation & Analysis:

The dimension of the intersection cannot exceed the dimension of either subspace; thus $\dim(W_1 \cap W_2) \le \min(3, 4) = 3$.

Common questions

Because every subspace must be a vector space in its own right, it must contain an additive identity (zero vector). If a subset fails to contain the zero vector, checking closure axioms is unnecessary as it is disqualified instantly.

Candidates often confuse the set of polynomials of exact degree $n$ with polynomials of degree at most $n$ ($P_n$). Exact degree $n$ polynomials do not form a subspace because the sum of two polynomials can result in cancellation of the highest degree term.

Use elimination heuristics: immediately reject subsets with non-zero constants in linear equations, strict inequalities ($>, <, \ge$), or non-linear terms (like products $xy$ or degrees).

← Previous Topic

Transpose of a linear transformation

Next Topic →

Basis of a vector space