7. Linear Algebra

Subspace

Q: FAQ 1: Why is the zero vector check so important for subspaces?

Because every vector space must contain an additive identity (zero vector). If a subset does not contain the zero vector of the parent space, it automatically violates the vector space axioms and cannot be a subspace.

Q: FAQ 2: What is the most common reason students lose marks on subspace questions?

Falling for affine subsets (like $Ax = b$ where $b \neq 0$ or lines not passing through the origin) or assuming unions of subspaces are subspaces.

Q: FAQ 3: How can candidates solve subspace questions faster during the exam?

Use elimination shortcuts: immediately test for the zero vector and check whether defining equations are homogeneous and linear.

A subspace is a subset of a vector space that is itself a vector space under the inherited operations of addition and scalar multiplication.

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

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Definition and statement

Fundamental Subspace Characterization Theorem

Let $V$ be a vector space over a field $\mathbb{F}$, and let $W$ be a non-empty subset of $V$. Then $W$ is a subspace of $V$ if and only if for all $\alpha, \beta \in \mathbb{F}$ and for all $x, y \in \mathbb{W}$, we have $\alpha x + \beta y \in W$.

📐 Geometric Interpretation

Geometrically, a subspace represents a flat geometric object passing through the origin (such as lines through the origin in $\mathbb{R}^2$, or planes and lines through the origin in $\mathbb{R}^3$). It can never be a translated affine space that misses the zero vector.

⚙️ Algebraic & Structural Meaning

It condenses the eight vector space axioms down into closedness under linear combinations, ensuring that the algebraic operations do not take vectors outside the subset.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Non-emptiness ($W \neq \emptyset$)	`Set membership condition`	Strictly Mandatory / Necessary	The empty set fails to contain the zero vector and cannot satisfy any vector space axioms.
Closure under addition ($x + y \in W$)	`Binary operation on $W$`	Strictly Mandatory	If vector addition leads outside $W$, the set fails to be closed under vector addition.
Closure under scalar multiplication ($cx \in W$)	`External multiplication`	Essential for conclusion	If scaling a vector pulls it out of $W$ (e.g., restricted positive quadrants), scaling by negative numbers or fractions fails.

Useful results and shortcuts

Use these identities and results when their conditions apply:

The Zero Vector Quick-Test

$$ 0_V \notin W \implies W \text{ is NOT a subspace} $$

Rule: Always check if the zero vector of the parent space belongs to the given subset $W$. If $0$ is missing, immediately reject the option without checking closure.

Application: Frequently used in UP PGT/TGT to eliminate 2 out of 4 options.

Homogeneous vs Non-Homogeneous Equations

$$ A\mathbf{x} = \mathbf{0} \implies \text{Subspace} $$

Rule: Solution sets of homogeneous linear systems are always subspaces of $\mathbb{R}^n$, whereas non-homogeneous systems are affine varieties and never subspaces.

Non-Linearity Elimination

$$ x^2 + y^2 \le 1 \text{ or } xy = 0 \implies \text{Not a Subspace} $$

Rule: Subspaces are strictly defined by linear homogeneous equations. Any degree $\ge 2$ terms, products of variables ($xy$), or non-linear constraints instantly rule out a subspace.

Intersection of Subspaces

$$ W_1, W_2 \le V \implies W_1 \cap W_2 \le V $$

Rule: The intersection of any collection of subspaces is always a subspace. (Note: Union is generally not a subspace unless one is contained in the other).

Application: Tested as theoretical assertion-reason questions in PGT screening tests.

Dimension Constraint of Subspaces

$$ W \le V \implies \dim(W) \le \dim(V) $$

Rule: If a purported subspace has a dimension greater than the parent space, or fails to satisfy rank-nullity bounds, it is invalid.

Application: Applied in vector space mapping and matrix subspace problems.

Symmetric and Skew-Symmetric Matrices

$$ \text{Sym}_n(\mathbb{R}) \le M_n(\mathbb{R}), \quad \text{Skew}_n(\mathbb{R}) \le M_n(\mathbb{R}) $$

Rule: Symmetric matrices form a subspace of $M_n(\mathbb{R})$, and skew-symmetric matrices form a subspace. However, their union or general invertible matrices do not.

Application: High-frequency question type in UP TGT/PGT linear algebra.

Common errors and counterexamples

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

⚠️ COMMON ERROR #01

The Zero Vector Fallacy in Transformed Sets

Common Pitfall: Candidates check addition and scalar multiplication but forget that $0_V$ must be in the subset.

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

Why it fails: Here $(0,0)$ does not satisfy $0 + 0 = 1$. Thus $0_V \notin W$, so $W$ is not a subspace despite looking linear.

🛡️ Exam Defense: Substitute $(0,0, \dots, 0)$ into the defining equation first before doing any closure tests.

⚠️ COMMON ERROR #02

Union of Subspaces Confusion

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

Counterexample: W_1 = \{(x, 0) \mid x \in \mathbb{R}\}, \; W_2 = \{(0, y) \mid y \in \mathbb{R}\} \text{ in } \mathbb{R}^2

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

🛡️ Exam Defense: Remember: $W_1 \cup W_2$ is a subspace $\iff W_1 \subset W_2$ or $W_2 \subset W_1$.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Subspace Test	Non-empty subset $W \subseteq V$	$\alpha x + \beta y \in W$	Flat passing through origin	Linear homogeneous equations
Intersection of Subspaces	$W_1, W_2 \le V$	$W_1 \cap W_2 \le V$	Common intersection slice	Always forms a subspace
Union of Subspaces	$W_1, W_2 \le V$	$W_1 \cup W_2 \le V \iff W_1 \subseteq W_2$ or $W_2 \subseteq W_1$	Combined planar regions	Usually NOT a subspace
Sum of Subspaces	$W_1, W_2 \le V$	$W_1 + W_2 = \{w_1 + w_2 \mid w_i \in W_i\}$	Minkowski sum of sets	Forms a subspace

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 x = y = z\}(C) \{(x, y, z) \mid xyz = 0\}(D) \{(x, y, z) \mid x \ge 0\}

📚 Method A: Formal Textbook Derivation

Step 1: Check zero vector. For (B), $0=0=0$ holds, so $(0,0,0) \in W$. Step 2: Check closure: if $x_1=y_1=z_1$ and $x_2=y_2=z_2$, then $(x_1+x_2)=(y_1+y_2)=(z_1+z_2)$, which holds. Thus (B) is a subspace.

⚡ Method B: 25-Second Exam Shortcut

Option (A) misses $(0,0,0)$ because $0+0+0 \neq 1$. Option (C) has non-linear product $xyz$. Option (D) fails scalar multiplication by negative numbers. Only (B) represents a homogeneous linear system.

💡 Tip: Look for homogeneous linear equations to instantly spot the correct subspace.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Let $V = M_n(\mathbb{R})$. Which of the following is NOT a subspace of $V$?

(A) Diagonal matrices(B) Symmetric matrices(C) Invertible matrices(D) Upper triangular matrices

📚 Method A: Formal Textbook Derivation

The zero matrix $0$ is not invertible, hence the zero vector of $M_n(\mathbb{R})$ does not belong to the set of invertible matrices. Also, the sum of two invertible matrices may be singular.

⚡ Method B: 25-Second Exam Shortcut

Recall the golden rule: Invertible matrices do NOT contain the zero matrix, so they can never form a subspace.

💡 Tip: Examiners frequently include 'invertible matrices' or 'singular matrices' as distractors.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: Which of the following subsets of $\mathbb{R}^2$ is a subspace?

A $\{(x, y) \mid y = 2x\}$

B $\{(x, y) \mid y = 2x + 1\}$

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

D $\{(x, y) \mid x^2 + y^2 \le 1\}$

Derivation & Analysis:

$y = 2x$ passes through the origin ($0,0$) and represents a homogeneous linear equation, defining a line through the origin which is a subspace of $\mathbb{R}^2$.

MCQ 2: 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:

Standard theorem in linear algebra: The union of two subspaces is a subspace if and only if one is contained in the other.

MCQ 3: The dimension of the subspace of all symmetric matrices of order $n$ over $\mathbb{R}$ is:

A $n^2$

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

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

D $n^2 - n$

Derivation & Analysis:

A symmetric matrix has independent entries on and above the main diagonal, numbering $1 + 2 + \dots + n = \frac{n(n+1)}{2}$.

MCQ 4: The dimension of the subspace of all skew-symmetric matrices of order $n$ over $\mathbb{R}$ is:

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

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

C $n^2 - n$

D $\frac{n^2}{2}$

Derivation & Analysis:

Skew-symmetric matrices have zeros on the main diagonal and negated entries below, giving $\frac{n(n-1)}{2}$ independent parameters.

MCQ 5: Which of the following is a subspace of the vector space $\mathbb{R}^{\mathbb{R}}$ of all real-valued functions?

A $\{f \mid f(0) = 1\}$

B $\{f \mid f(1) = 0\}$

C $\{f \mid f is differentiable and f'(x) = 1\}$

D $\{f \mid f(x) \ge 0 \forall x\}$

Derivation & Analysis:

The zero function $f(x)=0$ satisfies $f(1)=0$, and linear combinations of functions vanishing at $x=1$ also vanish at $x=1$. Thus it forms a subspace.

MCQ 6: If $W_1$ and $W_2$ are two subspaces of dimension $3$ and $5$ respectively in a vector space of dimension $7$, what is the minimum possible dimension of $W_1 \cap W_2$?

A $0$

B $1$

C $2$

D $3$

Derivation & Analysis:

Using modular identity for dimensions: $\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2)$. Since $\dim(W_1 + W_2) \le 7$, we have $3 + 5 - \dim(W_1 \cap W_2) \le 7 \implies \dim(W_1 \cap W_2) \ge 1$.

MCQ 7: The set of all polynomials of degree at exact degree $n$ along with the zero polynomial:

A Forms a subspace of $P(\mathbb{R})$

B Does not form a subspace because of closure under addition failure

C Forms a subspace of dimension $n$

D Forms a subspace of dimension $n+1$

Derivation & Analysis:

The sum of polynomials of exact degree $n$, such as $(x^n + x) + (-x^n + 1) = x + 1$, has degree $1 < n$, failing addition closure.

MCQ 8: Consider the subset $W$ of $\mathbb{R}^3$ given by $x - 2y + 3z = 0$. The dimension of $W$ is:

A $0$

B $1$

C $2$

D $3$

Derivation & Analysis:

The equation $x - 2y + 3z = 0$ is 1 independent homogeneous linear equation in 3 variables, so $\dim(W) = 3 - 1 = 2$.

MCQ 9: Which of the following is a subspace of the vector space of all $n \times n$ matrices?

A Matrices with trace equal to $1$

B Matrices with determinant equal to $0$

C Matrices with trace equal to $0$

D Orthogonal matrices

Derivation & Analysis:

Trace is a linear map, and the kernel of a linear map (matrices with trace zero) is always a subspace.

Q10

MCQ 10: Let $W$ be the set of all symmetric matrices of order $n$. What is the codimension of $W$ in $M_n(\mathbb{R})$?

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

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

C $n^2$

D $\frac{n(n-2)}{2}$

Derivation & Analysis:

Codimension is $\dim(M_n(\mathbb{R})) - \dim(\text{Sym}_n) = n^2 - \frac{n(n+1)}{2} = \frac{n(n-1)}{2}$.

Common questions

Because every vector space must contain an additive identity (zero vector). If a subset does not contain the zero vector of the parent space, it automatically violates the vector space axioms and cannot be a subspace.

Falling for affine subsets (like $Ax = b$ where $b \neq 0$ or lines not passing through the origin) or assuming unions of subspaces are subspaces.

Use elimination shortcuts: immediately test for the zero vector and check whether defining equations are homogeneous and linear.