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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’ll 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’ll 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
- Classic examples: \(\mathbb{R}^n\), \(P_n\), \(M_{m\times n}(\mathbb{R})\), \(C[0,1]\)
Span, linear combinations, and solution spaces
- 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’re ready, return to the quiz at the top of the page and keep practicing vector spaces and subspaces.
& Subspaces
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
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
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.
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
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.
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
Summary
- Bases give unique coordinate representations.
- Dimension equals the number of vectors in a basis (and \(\dim V^*=\dim V\) for finite-dimensional \(V\)).
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
Summary
- \(U\cap W\) and \(U+W\) are always subspaces.
- \(U\cup W\) is usually not a subspace (unless one contains the other).
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’s 6 free parameters. So the dimension is \(6\).
Try it
Summary
- Dimensions count free parameters (degrees of freedom).
- Infinite-dimensional means no finite spanning set exists.
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
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.
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
Example: Compute \(\dim\{A\in M_{2\times3}(\mathbb{R}):A_{1,\ast}=0\}\).
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 \(3\).
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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.
- Basis: spans + independent; gives unique coordinates.
- Dimension: number of basis vectors; counts degrees of freedom.
- Operations: \(U\cap W\) and \(U+W\) are subspaces; \(U\cup W\) usually isn’t.
- 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).