Questions: Dimension of Vector Spaces

Question 1

Consider the vector space of all 2×2 matrices with real entries. What is its dimension?

• A. 2, because matrices are 2-dimensional arrays
• B. 4, because a basis consists of the four matrices with a single 1 and three 0s
• C. Infinite, because there are infinitely many possible matrices
• D. 2, because the matrix has 2 rows and 2 columns

Answer: (B)

Dimension is the number of vectors in any basis — not the 'size' of objects in the space. The standard basis for 2×2 matrices consists of the four matrices E₁₁, E₁₂, E₂₁, E₂₂ (each with a 1 in one position and 0s elsewhere), so the dimension is 4. This space is isomorphic to ℝ⁴, even though it looks different. The common error is confusing the dimensions of the objects (2×2 arrays) with the dimension of the space they form.

Question 2

A subspace W of ℝ⁵ has a basis {v₁, v₂, v₃}. A student claims W might also have a basis with 4 vectors. Is this possible?

• A. Yes — different bases of a space can have different numbers of vectors
• B. No — the exchange lemma guarantees all bases of a vector space have the same cardinality
• C. Only if the 4-vector set is linearly independent but does not span W
• D. Yes — it depends on the field over which the space is defined

Answer: (B)

The exchange lemma is precisely what guarantees dimension is well-defined: any two bases of the same vector space must have the same number of vectors. If W has a basis of 3 vectors, every basis of W has exactly 3 vectors — there is no way to construct a basis of 4. A set of 4 vectors that spans W must be linearly dependent (redundant), so it contains a smaller spanning subset; a set of 4 linearly independent vectors in W cannot span it if dim(W) = 3.

Question 3

Any two vector spaces over the same field with the same dimension are structurally identical — meaning they are isomorphic.

• T. True
• F. False

Answer: True

Dimension is the complete invariant for finite-dimensional vector spaces: knowing the dimension and the base field is enough to classify the space up to isomorphism. ℝ⁴ and the space of 2×2 real matrices have the same dimension (4) and the same field (ℝ), so they are isomorphic — there is a structure-preserving bijection between them. This is one reason dimension is such a powerful concept: it collapses an enormous diversity of seemingly different spaces into a single number.

Question 4

The dimension of a vector space can vary depending on which basis you choose to measure it from.

• T. True
• F. False

Answer: False

This is the misconception that the exchange lemma directly refutes. Dimension is well-defined precisely because all bases have the same cardinality — you can compute dim(V) from any basis and always get the same answer. If B₁ and B₂ are both bases of V, then |B₁| = |B₂| without exception. 'Dimension depends on basis choice' would make the concept meaningless; the whole point is that it is an intrinsic property of the space, not of any particular basis.

Question 5

Why is it significant that all bases of a vector space have the same number of vectors? What would break if this were not guaranteed?

Model answer: If different bases could have different sizes, 'dimension' would not be well-defined — it would depend on which basis you chose to count, making it useless as a property of the space itself. The invariance of basis size (guaranteed by the exchange lemma) is what makes dimension an intrinsic property of the vector space, independent of any particular coordinate system. Without this, the rank-nullity theorem, the classification of vector spaces by dimension, and dimensional arguments throughout linear algebra would all collapse.

The well-definedness of dimension is foundational. Rank of a matrix is defined as the dimension of the column space — if dimension depended on basis choice, rank would be ambiguous. The rank-nullity theorem states rank + nullity = n; this conservation law only makes sense if rank and nullity are fixed numbers, not basis-dependent quantities. The exchange lemma is the load-bearing result that makes everything else stand.