Vector Spaces & Linear Independence Reference Cheat Sheet

Vector spaces are the basic setting for linear algebra, where vectors can be added and scaled in a consistent way. This cheat sheet helps students recognize vector spaces, test subspaces, and understand how vectors generate larger sets. It is useful for organizing definitions, theorem statements, and common proof patterns in one printable reference.

The most important ideas are span, linear independence, basis, dimension, and linear transformations. A set spans a space when every vector in the space can be written as a linear combination of the set. A basis is both spanning and linearly independent, so every vector has a unique coordinate representation.

Rank-nullity connects the image and kernel of a linear map through $\dim(V) = \operatorname{rank}(T) + \operatorname{nullity}(T)$ .

Key Facts

A vector space $V$ over a field $\mathbb{F}$ is closed under vector addition and scalar multiplication and satisfies the standard addition, scalar, identity, and inverse axioms.
A nonempty subset $W \subseteq V$ is a subspace if for all $u,v \in W$ and $c \in \mathbb{F}$ , $u+v \in W$ and $cu \in W$ .
The span of vectors $v_1,\dots,v_k$ is $\operatorname{span}\{v_1,\dots,v_k\}=\{c_1v_1+\cdots+c_kv_k: c_i \in \mathbb{F}\}$ .
Vectors $v_1,\dots,v_k$ are linearly independent if $c_1v_1+\cdots+c_kv_k=0$ implies $c_1=\cdots=c_k=0$ .
Vectors $v_1,\dots,v_k$ are linearly dependent if there is a nontrivial solution to $c_1v_1+\cdots+c_kv_k=0$ .
A basis for $V$ is a linearly independent set that spans $V$ , and every vector $v \in V$ has a unique form $v=c_1b_1+\cdots+c_nb_n$ .
The dimension of a finite-dimensional vector space is the number of vectors in any basis, written $\dim(V)=n$ .
For a linear transformation $T:V\to W$ , the rank-nullity theorem is $\dim(V)=\operatorname{rank}(T)+\operatorname{nullity}(T)$ .

Vocabulary

Vector Space: A set $V$ with vector addition and scalar multiplication that satisfies the vector space axioms over a field $\mathbb{F}$ .
Subspace: A subset $W$ of a vector space $V$ that is itself a vector space under the same operations.
Span: The set of all linear combinations of given vectors, written $\operatorname{span}\{v_1,\dots,v_k\}$ .
Linear Independence: A property of vectors where the only solution to $c_1v_1+\cdots+c_kv_k=0$ is the trivial solution $c_1=\cdots=c_k=0$ .
Basis: A set of vectors that spans a vector space and is linearly independent.
Dimension: The number of vectors in any basis of a finite-dimensional vector space.

Common Mistakes to Avoid

Forgetting to check the zero vector in a subspace test is wrong because every subspace must contain $0$ as a consequence of closure under scalar multiplication.
Testing linear independence by checking only whether vectors are nonzero is wrong because nonzero vectors can still satisfy a nontrivial equation such as $c_1v_1+c_2v_2=0$ .
Assuming a spanning set is automatically a basis is wrong because a basis must also be linearly independent.
Confusing rank and nullity is wrong because $\operatorname{rank}(T)$ measures the dimension of the image while $\operatorname{nullity}(T)$ measures the dimension of the kernel.
Using row operations on column vectors without preserving the correct matrix interpretation is wrong because linear dependence of columns is determined by solving $Ax=0$ .

Practice Questions

1 Determine whether the vectors $(1,2,3)$ , $(2,4,6)$ , and $(0,1,1)$ in $\mathbb{R}^3$ are linearly independent.
2 Find a basis and the dimension of the subspace $W=\{(x,y,z)\in\mathbb{R}^3: x+y+z=0\}$ .
3 For a linear map $T:\mathbb{R}^5\to\mathbb{R}^3$ with $\operatorname{rank}(T)=2$ , find $\operatorname{nullity}(T)$ .
4 Explain why a set of four vectors in $\mathbb{R}^3$ cannot be linearly independent, even if none of the vectors is the zero vector.

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Vector Spaces & Linear Independence Reference Cheat Sheet

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