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Linear Algebra Fundamentals
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Linear Algebra Fundamentals
📊 Part 1: Matrix Operations
1. Given matrices A = [2 1; 3 4] and B = [1 2; 0 3], calculate A + B:
2. For the same matrices A and B, find AB (matrix multiplication):
3. Which condition must be satisfied for a matrix to be invertible?
The matrix must be square
The determinant must be non-zero
All entries must be positive
The matrix must be symmetric
4. Calculate the determinant of matrix C = [2 1; 5 3]:
⚖️ Part 2: Linear Systems
5. Solve the system of equations using Gaussian elimination:
2x + y = 7
x + 3y = 8
2x + y = 7
x + 3y = 8
6. For which type of linear system would you prefer to use the inverse matrix method rather than Gaussian elimination?
7. A bakery produces three types of bread. Let x = white loaves, y = brown loaves, z = wholemeal loaves.
The constraints are: x + y + z = 100, 2x + y = 120, x + 3z = 90
Set up this system in matrix form Ax = b:
The constraints are: x + y + z = 100, 2x + y = 120, x + 3z = 90
Set up this system in matrix form Ax = b:
📐 Part 3: Vector Spaces
8. Which of the following sets form a subspace of ℝ³? (Select all that apply)
All vectors of the form (a, 0, b) where a, b ∈ ℝ
All vectors of the form (a, 1, b) where a, b ∈ ℝ
All vectors (x, y, z) where x + y + z = 0
All vectors with positive components only
9. Determine whether the vectors v₁ = (1, 2, 0), v₂ = (0, 1, 1), and v₃ = (1, 3, 1) are linearly independent. Show your working:
10. Explain why a matrix must be invertible to use the inverse method to solve a system of equations:
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