How unit 4 is examined
This unit covers vector spaces, subspaces, linear combination, dependence and independence, basis and linear transformations; the subspace test and the independence proof are the parts that have carried marks.
Vector Space
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Definition. ==A vector space over a field $F$ is a non-empty set $V$ with addition and scalar multiplication such that $V$ is an abelian group under addition and, for all $a,b\in F$ and $u,v\in V$, $a(u+v)=au+av$, $(a+b)u=au+bu$, $(ab)u=a(bu)$ and $1u=u$.==
Key points.
- Closure: $u+v\in V$ and $au\in V$ for every $u,v\in V$, $a\in F$.
- Addition is commutative and associative, has a zero vector $0$, and every $u$ has a negative $-u$.
- Standard examples are $\mathbb{R}^n$, the set of $m\times n$ matrices and the set of polynomials.
Vector Sub Space
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Definition. <mark>A non-empty subset $W$ of a vector space $V(F)$ is a subspace if $W$ is itself a vector space under the same operations.</mark>
Key points.
- Test: $W$ is a subspace if $0\in W$, and $u,v\in W\Rightarrow u+v\in W$, and $u\in W,\ k\in F\Rightarrow ku\in W$.
- Equivalently, $au+bv\in W$ for all $u,v\in W$ and scalars $a,b$.
- Every space has the trivial subspaces $\{0\}$ and $V$.
- Example. For $W=\{(a,b,0)\}$: $(0,0,0)\in W$; $(a,b,0)+(c,d,0)=(a+c,b+d,0)\in W$; $k(a,b,0)=(ka,kb,0)\in W$. So $W$ is a subspace of $\mathbb{R}^3$.
Asked: [7 marks] (Dec 2024) Show that the set $W=\{(a,b,0): a,b\in\mathbb{R}\}$ is a subspace of $\mathbb{R}^3$.
Linear Combination of Vectors
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Definition. ==A vector $v$ is a linear combination of $v_1,\dots,v_n$ if $v=c_1v_1+c_2v_2+\dots+c_nv_n$ for some scalars $c_i$.==
Key points.
- To test whether $v$ is a combination, solve for the $c_i$; if the system is consistent, it is.
- The set of all combinations of $v_1,\dots,v_n$ is a subspace called their span.
Linearly Dependent
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Definition. ==Vectors $v_1,\dots,v_n$ are linearly dependent if $c_1v_1+\dots+c_nv_n=0$ holds with at least one $c_i\neq 0$.==
Key points.
- Then some vector is a linear combination of the others.
- Any set containing the zero vector is dependent.
- In $\mathbb{R}^n$, more than $n$ vectors are always dependent.
Linearly Independent
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Definition. ==Vectors $v_1,\dots,v_n$ are linearly independent if $c_1v_1+\dots+c_nv_n=0$ forces every $c_i=0$.==
Key points.
- Put the vectors as rows of a matrix; they are independent exactly when its rank equals the number of vectors.
- Proof for the question. Let $aw_1+bw_2=0$. Then $aw_1=-bw_2\in W_1\cap W_2=\{0\}$, so $aw_1=0$; as $w_1\neq0$, $a=0$. Likewise $bw_2=0$ gives $b=0$. Hence $\{w_1,w_2\}$ is independent.
Asked: [7 marks] (Jun 2025) Let $W_1,W_2$ be subspaces of $V$ with $W_1\cap W_2=\{0\}$, and $w_1\in W_1$, $w_2\in W_2$ nonzero. Prove $\{w_1,w_2\}$ is linearly independent.
Basis of a Vector Space
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Definition. <mark>A basis of $V$ is a linearly independent set of vectors that spans $V$.</mark>
Key points.
- Every vector of $V$ is a unique combination of the basis vectors.
- All bases of $V$ have the same number of vectors, called the dimension of $V$.
- $\mathbb{R}^n$ has the standard basis $e_1,\dots,e_n$ and dimension $n$.
Linear Transformations
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Definition. ==A map $T:V\to W$ is a linear transformation if $T(u+v)=T(u)+T(v)$ and $T(ku)=kT(u)$ for all $u,v\in V$ and scalars $k$.==
Key points.
- Equivalently, $T(au+bv)=aT(u)+bT(v)$.
- Always $T(0)=0$.
- Kernel $=\{v:T(v)=0\}$ and range $=\{T(v)\}$ are subspaces, with $\dim V=\dim\ker T+\dim\operatorname{range}T$.
Last-minute revision
- Subspace test: contains $0$, closed under addition and scalar multiplication.
- Linear combination: $v=\sum c_iv_i$; the span of vectors is a subspace.
- Dependent: a nontrivial combination equals $0$; independent: only the trivial one does.
- Any set with the zero vector is dependent; more than $n$ vectors in $\mathbb{R}^n$ are dependent.
- Basis = independent + spanning; the number of basis vectors is the dimension.
- $\dim\mathbb{R}^n=n$.
- Linear map: $T(au+bv)=aT(u)+bT(v)$, and $T(0)=0$.
- Rank-nullity: $\dim V=\dim\ker T+\dim\operatorname{range}T$.
Memory hooks
- Subspace test: "Zero, Add, Scale".
- Basis = "independent AND spanning": no waste, no gaps.
- Dependent means one vector is redundant.
Coverage checklist
- Vector Space: definition and axioms.
- Vector Sub Space: Dec 2024 subspace $\{(a,b,0)\}$ of $\mathbb{R}^3$.
- Linear Combination of Vectors: definition and span.
- Linearly Dependent: definition and properties.
- Linearly Independent: Jun 2025 proof for $\{w_1,w_2\}$.
- Basis of a Vector Space: definition, dimension.
- Linear Transformations: definition, kernel and range.