Vector Spaces & Subspaces Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page to practice vector spaces and subspaces - the foundation of Linear Algebra: vector space axioms (closure, associativity, distributivity, identity, inverses), the fast subspace test (contains \(0\), closed under addition and scalar multiplication), linear combinations and span, basis and dimension, coordinates relative to a basis (change of basis), standard subspaces like null space and solution spaces, sum and intersection of subspaces (\(U+W\) and \(U\cap W\)), and the meaning of quotient spaces \(V/W\). You will also see key examples in \(\mathbb{R}^n\), matrix spaces \(M_{m\times n}(\mathbb{R})\), polynomial spaces \(P_n\), and function spaces like \(C[0,1]\). If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
How this vector spaces and subspaces practice works
1. Take the quiz: answer the vector space, subspace, span, basis, and dimension questions at the top of the page.
2. Open the lesson (optional): review vector space axioms, the subspace test, spans, bases, coordinates, dimension, and quotient spaces with clear examples.
3. Retry: return to the quiz and apply the subspace test and basis/dimension tools immediately.
What you will learn in the vector spaces & subspaces lesson
Vector spaces & the subspace test
Vector space definition: operations + axioms (including additive identity \(0\))
Subspace test: \(0\in U\), closed under addition and scalar multiplication
Span as all linear combinations: \(\text{span}\{v_1,\dots,v_k\}\)
Solution spaces of homogeneous systems \(Ax=0\) are subspaces
Null space and column space as core subspaces in linear algebra
Basis, coordinates, and dimension
Basis: spanning + linear independence
Coordinates relative to a basis (change of basis computations)
Dimension: size of a basis; compute dimensions of common subspaces
Subspace operations & quotient spaces
Intersection \(U\cap W\) is always a subspace
Sum \(U+W\) is the smallest subspace containing both \(U\) and \(W\)
Quotient space \(V/W\): vectors modulo the subspace \(W\) (cosets)
Back to the quiz
When you are ready, return to the quiz at the top of the page and keep practicing vector spaces and subspaces.
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Vector Spaces & Subspaces
Step-by-Step Guide
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Vector Spaces & Subspaces Lesson
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Lesson Overview
Lesson overview
Purpose: Build a clear understanding of vector spaces and subspaces so you can use the subspace test, describe sets using span and linear combinations, compute bases, coordinates relative to a basis, and dimension, recognize solution spaces (null spaces) as subspaces, and understand sum, intersection, and quotient spaces \(V/W\) via cosets.
Success criteria
State what a vector space is (operations + axioms) and identify the zero vector.
Use the subspace test: verify \(0\in U\), closure under addition, and closure under scalar multiplication.
Describe a set as a span of vectors and interpret linear combinations.
Recognize that solution spaces of homogeneous linear systems \(Ax=0\) are subspaces.
Find and interpret a basis (spanning + linear independence) and compute dimension.
Compute coordinates relative to a basis (change of basis in \(\mathbb{R}^n\)).
Work with sum \(U+W\) and intersection \(U\cap W\) of subspaces.
Interpret the quotient space \(V/W\) as the set of cosets \(v+W\).
Use common examples: \(\mathbb{R}^n\), matrix spaces \(M_{m\times n}(\mathbb{R})\), polynomial spaces \(P_n\), and function spaces like \(C[0,1]\).
Key vocabulary
Vector space: a set \(V\) with addition and scalar multiplication satisfying the axioms (including an additive identity \(0\)).
Subspace: a subset \(U\subseteq V\) that is itself a vector space with the inherited operations.
Span: \(\text{span}(S)\) is the set of all linear combinations of vectors in \(S\).
Basis: a linearly independent set that spans the space; every vector has a unique coordinate representation.
Dimension: the number of vectors in any basis of a finite-dimensional vector space.
Coset / quotient space: \(v+W=\{v+w:w\in W\}\); \(V/W\) is the set of all such cosets.
Quick pre-check
Pre-check 1: Which vector is always present in any subspace of a vector space \(V\)?
Hint: A subspace must contain the additive identity.
Pre-check 2: If \(U\) and \(W\) are subspaces of \(V\), what is always true about \(U \cap W\)?
Hint: Check the subspace test for the intersection.
Vector Spaces & Subspaces
Vector spaces, subspaces, and the subspace test
Learning goal: Decide quickly whether a set is a subspace using a small checklist, and avoid common traps (missing \(0\), not closed under addition, not closed under scalar multiplication).
Key idea
A vector space \(V\) over a field (like \(\mathbb{R}\)) is a set with two operations: vector addition and scalar multiplication, satisfying the standard axioms (associativity, commutativity of addition, distributive laws, scalar identity, additive identity \(0\), additive inverses, and closure).
A subset \(U\subseteq V\) is a subspace if it is a vector space using the same operations. In practice, use the subspace test.
The subspace test
Zero vector: \(0 \in U\).
Closed under addition: if \(u,v\in U\), then \(u+v\in U\).
Closed under scalar multiplication: if \(u\in U\) and \(c\in \mathbb{R}\), then \(cu\in U\).
Common examples
\(\mathbb{R}^n\) and any plane through the origin in \(\mathbb{R}^3\).
Matrix spaces like \(M_{m\times n}(\mathbb{R})\); sets defined by linear constraints are often subspaces.
Function spaces like \(C[0,1]\); sets defined by linear conditions (e.g. \(f(0)=0\)) are subspaces.
Worked example
Example: Is \(U=\{f\in C[0,1]: f(0)=0\}\) a subspace of \(C[0,1]\)?
Zero function: \(0(0)=0\), so \(0\in U\). If \(f(0)=0\) and \(g(0)=0\), then \((f+g)(0)=f(0)+g(0)=0\), so \(f+g\in U\). If \(f(0)=0\) and \(c\in\mathbb{R}\), then \((cf)(0)=c f(0)=0\), so \(cf\in U\). Therefore \(U\) is a subspace.
Try it
Try it 1: Is the set \(S=\{(x,y)\in\mathbb{R}^2: x+y=1\}\) a subspace of \(\mathbb{R}^2\)?
Hint: Plug \((0,0)\) into \(x+y=1\).
Try it 2: If \(w\) is in a subspace \(S\), what can you say about \(-2w\)?
Hint: A subspace is closed under multiplying by any scalar in the field.
Summary
A subspace must contain \(0\) and be closed under addition and scalar multiplication.
Sets like \(x+y=1\) are usually affine (shifted) and fail the \(0\in U\) test.
Span & Linear Combinations
Linear combinations, span, and describing subspaces
Learning goal: Translate between “all linear combinations” and a clean subspace description, and recognize solution spaces as subspaces.
Key idea
A linear combination of vectors \(v_1,\dots,v_k\) is any vector of the form \[ c_1 v_1 + \cdots + c_k v_k \] where \(c_1,\dots,c_k\in\mathbb{R}\). The span is the set of all such combinations: \[ \text{span}\{v_1,\dots,v_k\}=\{c_1 v_1 + \cdots + c_k v_k : c_i\in\mathbb{R}\}. \] A span is always a subspace.
Solution spaces are subspaces
The solution set of a homogeneous system \(Ax=0\) is the null space of \(A\), \[ \mathcal{N}(A)=\{x: Ax=0\}, \] and it is always a subspace of \(\mathbb{R}^n\).
Worked example
Example: Find the dimension of the solution space of \(x+y+z=0\) in \(\mathbb{R}^3\).
Solve \(x+y+z=0\) by expressing one variable in terms of the others: \(x=-y-z\). Let \(y=s\) and \(z=t\). Then \[ (x,y,z)=(-s-t,\, s,\, t)=s(-1,1,0)+t(-1,0,1). \] The solution space is \(\text{span}\{(-1,1,0),(-1,0,1)\}\), which is a plane through the origin, so its dimension is \(2\).
Try it
Try it 1: What is the null space of the differentiation map \(D: P_2 \to P_1\)?
Hint: \(Dp=0\) means the polynomial has zero derivative.
Try it 2: What is the dimension of \(\{A\in M_{2\times3}(\mathbb{R}): A_{1,\ast}=0\}\)?
Hint: A \(2\times3\) matrix has 6 entries; setting the entire first row to zero removes 3 degrees of freedom.
Summary
\(\text{span}(S)\) is always a subspace: it contains \(0\) and is closed under linear combinations.
Homogeneous solution spaces (null spaces) are subspaces and their dimension equals the number of free parameters.
Basis, Coordinates, Dimension
Bases, coordinates relative to a basis, and dimension
Learning goal: Use bases to represent vectors efficiently, compute coordinates, and connect basis size to dimension.
Key idea
A set \(B=\{b_1,\dots,b_n\}\) is a basis for a vector space \(V\) if: (1) it spans \(V\), and (2) it is linearly independent. When \(B\) is a basis, every \(v\in V\) can be written uniquely as \[ v = c_1 b_1 + \cdots + c_n b_n. \] The scalars \((c_1,\dots,c_n)\) are the coordinates of \(v\) relative to the basis \(B\).
Dimension
If \(V\) is finite-dimensional, the dimension \(\dim V\) is the number of vectors in any basis of \(V\). For example, \(\dim \mathbb{R}^n = n\), \(\dim P_n = n+1\), and \(\dim M_{m\times n}(\mathbb{R}) = mn\). If \(V\) has a basis of size \(5\), then \(\dim V = 5\) and \(\dim V^* = 5\) as well.
Worked example
Example: Relative to the basis \(B=\{(1,0),(1,1)\}\) of \(\mathbb{R}^2\), what are the coordinates of \((2,3)\)?
Write \((2,3)=a(1,0)+b(1,1)\). Then \((2,3)=(a+b,\, b)\), so \(b=3\) and \(a+b=2\Rightarrow a= -1\). Therefore the coordinate vector is \([\, (2,3)\, ]_B = \begin{pmatrix}-1\\3\end{pmatrix}\).
Try it
Try it 1: If \(V\) has a basis of size \(5\), what is \(\dim V^*\) (the dual space)?
Hint: For finite-dimensional vector spaces, \(\dim V = \dim V^*\).
Try it 2: Which subset of \(\mathbb{R}^3\) is a 2-dimensional subspace?
Hint: A 2D subspace in \(\mathbb{R}^3\) is typically a plane through the origin given by a homogeneous linear equation.
Summary
Bases give unique coordinate representations.
Dimension equals the number of vectors in a basis (and \(\dim V^*=\dim V\) for finite-dimensional \(V\)).
Subspace Operations
Intersection, sum, union, and common subspace geometry in \(\mathbb{R}^n\)
Learning goal: Work confidently with \(U\cap W\), \(U+W\), and understand why \(U\cup W\) is usually not a subspace.
Intersection \(U\cap W\)
If \(U\) and \(W\) are subspaces of \(V\), then \(U\cap W\) is always a subspace of \(V\). Reason: it contains \(0\) and is closed under addition/scalar multiplication because both \(U\) and \(W\) are.
Sum \(U+W\)
The sum of subspaces is \[ U+W=\{u+w: u\in U,\; w\in W\}. \] It is the smallest subspace containing both \(U\) and \(W\).
Union \(U\cup W\)
In general, the union of two subspaces is not a subspace. It is a subspace only in the special case where one subspace is contained in the other: \[ U\cup W \text{ is a subspace } \Longleftrightarrow U\subseteq W \text{ or } W\subseteq U. \]
Worked example
Example: What is the sum \(U+W\) of the \(x\text{–}y\) plane and the \(y\text{–}z\) plane in \(\mathbb{R}^3\)?
The \(x\text{–}y\) plane is \(U=\{(x,y,0)\}\). The \(y\text{–}z\) plane is \(W=\{(0,y,z)\}\). Add a general vector from each: \[ (x,y_1,0)+(0,y_2,z)=(x,\, y_1+y_2,\, z). \] This can produce any \((x,y,z)\in\mathbb{R}^3\), so \(U+W=\mathbb{R}^3\).
Try it
Try it 1: What is the intersection of subspaces \(\{(x,0,0)\}\) and \(\{(0,y,0)\}\) in \(\mathbb{R}^3\)?
Hint: A vector in the intersection must be on both axes at once.
Try it 2: When is the union of two subspaces \(U\) and \(W\) also a subspace?
Hint: If neither contains the other, pick \(u\in U\setminus W\) and \(w\in W\setminus U\) and check closure under addition.
Summary
\(U\cap W\) and \(U+W\) are always subspaces.
\(U\cup W\) is usually not a subspace (unless one contains the other).
Dimension & Infinite-Dimensional Spaces
Dimension in practice: constraints, degrees of freedom, and infinite dimension
Learning goal: Compute dimensions from free parameters or constraints, and recognize when a space is infinite-dimensional.
Key idea
Dimension is a “degrees of freedom” count. Every independent linear constraint typically reduces dimension by \(1\) (in finite-dimensional settings). For example, a single homogeneous linear equation in \(\mathbb{R}^3\) defines a 2D subspace (a plane through the origin).
Finite vs. infinite dimension
A vector space is finite-dimensional if it has a finite basis. If no finite basis exists, it is infinite-dimensional (for example, \(P\) = all polynomials, or \(C[0,1]\)). If a subspace \(S\) has infinite dimension, then it cannot be spanned by finitely many vectors (every spanning set must be infinite).
Worked example
Example: Is the set of all upper-triangular \(3\times 3\) matrices a subspace of \(M_{3\times 3}(\mathbb{R})\)? What is its dimension?
Upper-triangular matrices are closed under addition and scalar multiplication, and the zero matrix is upper-triangular, so it is a subspace. An upper-triangular \(3\times 3\) matrix has free entries in positions \((1,1),(1,2),(1,3),(2,2),(2,3),(3,3)\): that is 6 free parameters. So the dimension is \(6\).
Try it
Try it 1: If \(S\) is a subspace of infinite dimension, it must be:
Hint: Infinite-dimensional means no finite list of vectors can span the space.
Try it 2: Is the set \(\{0\}\) a subspace?
Hint: The zero vector alone satisfies the subspace test.
Summary
Dimensions count free parameters (degrees of freedom).
Infinite-dimensional means no finite spanning set exists.
Quotient Spaces
Quotient spaces \(V/W\): vectors modulo a subspace
Learning goal: Understand quotient spaces conceptually: “treat vectors that differ by something in \(W\) as the same.”
Key idea
Let \(W\) be a subspace of \(V\). Two vectors \(v\) and \(u\) are considered equivalent modulo \(W\) if \[ v-u \in W. \] The equivalence class of \(v\) is the coset \[ v+W = \{v+w : w\in W\}. \] The quotient space \(V/W\) is the set of all cosets: \[ V/W = \{v+W : v\in V\}. \]
How to think about \(V/W\)
You “collapse” the entire subspace \(W\) to act like the zero element in the quotient.
Vectors that differ by an element of \(W\) become the same point in \(V/W\).
This is useful for focusing on directions “not in \(W\)” and for simplifying structure.
Worked example
Example: Let \(V=\mathbb{R}^2\) and \(W=\text{span}\{(1,0)\}\) (the \(x\)-axis). What does \((0,3)+W\) look like?
\(W=\{(t,0): t\in\mathbb{R}\}\). Then \[ (0,3)+W=\{(0,3)+(t,0): t\in\mathbb{R}\}=\{(t,3): t\in\mathbb{R}\}, \] which is a horizontal line at height \(y=3\). In \(\mathbb{R}^2/W\), all points on that line represent the same coset.
Try it
Try it 1: Which describes the quotient space \(V/W\)?
Hint: Elements of \(V/W\) are not vectors of \(V\) but equivalence classes (cosets).
Try it 2: Can the intersection of two subspaces in \(\mathbb{R}^n\) be the zero subspace?
Hint: For example, the \(x\)-axis and \(y\)-axis in \(\mathbb{R}^2\) intersect only at \(0\).
Summary
\(V/W\) is the set of cosets \(v+W\), i.e. vectors modulo the subspace \(W\).
Quotients “collapse” directions in \(W\) so you focus on what remains.
Applications & Big Picture
Why vector spaces and subspaces matter
Learning goal: Connect the subspace viewpoint to the rest of linear algebra - and finish with a final check.
Where vector spaces and subspaces show up
Linear systems: solution sets of \(Ax=0\) are subspaces (null spaces).
Linear transformations: kernels and images are subspaces; dimension relates to rank and nullity.
Geometry: lines/planes through the origin are subspaces; sums and intersections match geometric intuition.
Data and ML: subspaces model low-dimensional structure inside high-dimensional data (PCA).
Functions: many spaces of functions are vector spaces; constraints like \(f(0)=0\) define subspaces.
Worked example: a clean dimension count in matrices
A \(2\times 3\) matrix has 6 entries. The condition \(A_{1,\ast}=0\) forces the entire first row to be zero, which fixes 3 entries. The second row \((a_{21},a_{22},a_{23})\) is free, giving 3 degrees of freedom. So the dimension is \(3\).
Try it
Try it 1: What is the dimension of the solution space of \(x + y + z = 0\) in \(\mathbb{R}^3\)?
Hint: One homogeneous linear equation in \(\mathbb{R}^3\) typically leaves two free parameters.
Try it 2: Is a subset of \(\mathbb{R}^n\) that does not include the zero vector a subspace?
Hint: The additive identity must be in every vector space (and hence every subspace).
Final recap
Subspace test: check \(0\in U\), closure under addition, closure under scalar multiplication.
Span: all linear combinations; \(\text{span}(S)\) is always a subspace.
Dimension: number of basis vectors; counts degrees of freedom.
Operations: \(U\cap W\) and \(U+W\) are subspaces; \(U\cup W\) usually is not.
Quotient: \(V/W\) is cosets \(v+W\), i.e. vectors modulo \(W\).
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the vector space or subspace skill you need (subspace test, span, basis/coordinates, dimension, sum/intersection, or quotient spaces).
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