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Section 1: Problems

University-level exam questions for Core Linear System Theory.

Conceptual Questions​

Question 1.1​

Can there be multiple bases for a given vector spaces?

Difficulty: Easy

Question 1.2​

If there are multiple bases, can the dimensions differ?

Difficulty: Easy

Question 1.3​

Prove: The rank rr of an m×nm \times n matrix is always less than or equal to both mm and nn. rank(A)≤min⁡(m,n)\text{rank}(A) \le \min(m, n)

Difficulty: Medium

Question 1.4​

If all columns of a matrix A are linearly independent, are there any element in its nullspace other than the zero vector? (i.e. dimension of N(A) > 0?)

Difficulty: Medium

Question 1.5​

Show that the reduced form of ATA^T is not the transpose of the reduced form of AA.

Difficulty: Medium

Vector Spaces​

Problem 2.1​

Determine whether the following sets form vector spaces:

a) All vectors in ℝ³ whose components sum to zero b) All matrices A such that A² = A c) All polynomials of degree exactly 3

Difficulty: Medium

Problem 2.2​

Let V be the space of all 2×2 matrices. Find a basis for V and verify it.

Difficulty: Medium

Problem 2.3​

Prove that the intersection of two subspaces is a subspace.

Difficulty: Hard

Linear Independence​

Problem 2.1​

Determine if the following vectors are linearly independent: v₁ = (1, 2, 3), v₂ = (2, 4, 6), v₃ = (1, 1, 1)

Difficulty: Easy

Problem 2.2​

Find the dimension of the span of 2

Difficulty: Medium

The Four Fundamental Subspaces​

Problem 3.1​

For matrix A = [[1, 2, 3], [2, 4, 6]], find:

  • Column space C(A)
  • Nullspace N(A)
  • Row space C(Aᵀ)
  • Left nullspace N(Aᵀ)

Difficulty: Medium

Problem 3.2​

Prove that dim(C(A)) + dim(N(Aᵀ)) = m for an m×n matrix A.

Difficulty: Hard

Linear Transformations​

Problem 4.1​

Let T: ℝ² → ℝ² be defined by T(x, y) = (2x - y, x + 3y). Find the matrix representation of T.

Difficulty: Easy

Problem 4.2​

Prove the rank-nullity theorem for a linear transformation T: V → W.

Difficulty: Hard

Matrix Operations​

Problem 5.1​

Compute the LU decomposition of: A = [[2, 1, 1], [4, 3, 3], [8, 7, 9]]

Difficulty: Medium

Problem 5.2​

Prove that (AB)ᵀ = BᵀAᵀ for any matrices A and B where AB is defined.

Difficulty: Medium

Challenge Problems​

Problem 6.1​

Let A be an n×n matrix. Prove that the four fundamental subspaces come in orthogonal pairs.

Difficulty: Very Hard

Problem 6.2​

Construct an example of a linear transformation that is injective but not surjective.

Difficulty: Hard

Solutions​

Solutions are available in the implementation file with verification code.