Linear Independence, Bases and Dimension

What does it mean for vectors to be Linearly Dependent?

A set \{v_1,…,v_q\}\subseteq\mathbb{F}^n is linearly dependent if there exists scalars, not all zero, such that \alpha_1v_1+\alpha_2v_2+…+\alpha_qv_Q=0. This means at least one vector can be expressed as a linear combination of the others.

What does it mean for vectors to be Linearly Independent?

Vectors \{v_1,…,v_q\} are linearly independent if \alpha_1v_1+…+\alpha_qv_q = 0\Rightarrow\alpha_1=…=\alpha_q=0. No vector in this set is redundant.

Why is any set containing the zero vector linearly dependent?

Because you can take the coefficient of the zero vector to be 1 and all others 0, and the linear combination will still equal zero.

What is the Span of a set of vectors?

The span of a set C\subseteq\mathb{F}^n, denoted Sp(C), is the set of all linear combinations of vectors from C. Sp(v_1,…v_q)=\{\sum^{q}_{j=1}\alpha_jv_j|\alpha_j\in\mathbb{F}\}

Is the span of any set of vectors a subspace?

Yes. The span of a collection of vectors is the smallest subspace containing all of them.

What does it mean for a set to span a subspace?

A set \{v_1,…v_q\} spans a subspace S\subseteq\mathbb{F}^n if every vector in S can be written as a linear combination of v_1,…,v_q\} i.e. S=Sp(v_1,…v_q).

What is the definition of a Basis?

A basis of a subspace S\ubseteq\mathbb{F}^n is a set of vectors that:

• Are linearly independent, and

• Span S

This means the basis uniquely represents every vector in the space via a linear combination.

What are the coordinates of a vector with respect to a basis?

Given a basis B=\{v_1,…,v_q\}, any u\in S can be uniquely written as u=\alpha_1v_1+…+\alpha_qv_q. The scalars \alpha_1,…,\alpha_q are the coordinates of u relative to B.

What is the Dimension of a subspace?

The dimension of a subspace S\subseteq\mathbb{F}^n, denoted dim(S), is the number of vectors in any basis of S. All bases of a subspace have the same size.

“Every subspace has a basis.” True or False?

True. Every subspace of \mathbb{F}^n has a basis, and every basis has the same number of elements (its dimension).

What is the Steinitz Exchange Lemma?

If \{v_1,…,v_q\} is a basis for a subspace and u\in S is not zero, then there exists some v_j such that replacing v_j with u yields another basis.

What are some of the Key Consequences of dimension?

If dim(S)=q then:

• Any linearly independent set in S has \leq q vectors.

• Any set of \lt q vectors is linearly dependent.

• Any set of q independent vectors is a basis.

• Any spanning set with \lt q vectors contains a basis as a subset

What is the dimension of \mathbb{F}^n?

The dimension is n, with the standard basis e_1=(1,0,…,0),…,e_n=(0,…,0,1).

What is a basis for the subspace \{(x_1,x_2,x_3)\in\mathbb{R}³ \ | \ x_1 + x_2 + x_3 = 0\}?

Vectors v_1=(1,0,-1), v_2=(0,1,-1) form a basis. The dimension is 2.

How do you check if a set of vectors forms a basis for a subspace?

1. Check linear independence (e.g. Solve Ax=0).

2. Check spanning (either directly or by comparing dimension).

If both hold, it’s a basis.