Linear Algebra for IEM
Complete Course Summary - University of Groningen - Block 1B
A student-compiled summary covering all midterm and final exam topics: systems of equations, matrix algebra,
determinants, eigenvalues & eigenvectors, vector spaces, geometry, and linear differential equations.
Topics: 1. Systems of Linear Equations | 2. Matrix Algebra | 3. Determinants | 4. Eigenvalues & Eigenvectors |
5. Vector Spaces & Subspaces | 6. Image, Kernel & Rank | 7. Linear Differential Equations | 8. Geometry in
Vector Spaces | 9. Linear Transformations
Linear Algebra for IEM | Page 1 of 8
, 1. Systems of Linear Equations
Compact form: Ax = b, where A is the coefficient matrix, x is the unknown vector, and b is the right-hand side.
No solution Unique solution Infinitely many
RREF gives a false row like 0 = 3. rank(A) = n. Every variable is a At least one free variable (column
pivot variable. without a pivot).
Row Operations
* Eij(d): add d x (row j) to row i
* Ei(c): multiply row i by scalar c (c not 0)
* Eij: swap rows i and j
Gaussian elimination -> RRF (zeros below pivots). Gauss-Jordan -> RREF (leading 1s, zeros above AND below
pivots).
LU Factorisation
A = LU
L = unit lower-triangular (1s on diagonal, Gauss multipliers below).
U = upper-triangular result after forward elimination.
det(A) = det(L) x det(U) = 1 x (product of U diagonal entries).
How to build L: start with the identity matrix and insert the multiplier mij (with original sign) at position (i, j).
Rank and Consistency
rank(A) = number of pivot columns = dim(col space) = dim(row space).
* rank(A) = n -> unique solution (no free variables).
* rank(A) < n -> infinitely many solutions.
* rank([A|b]) > rank(A) -> inconsistent, no solution.
2. Matrix Algebra
Matrix product AB: defined only when cols(A) = rows(B). Result is m x p when A is m x n and B is n x p.
NOTE: AB != BA in general. AB = AC does NOT imply B = C.
Key Identities
Transpose rules (AT)T = A | (AB)T = BTAT | (A+B)T = AT+BT
Inverse (2x2) A = [a b; c d] -> A-1 = 1/(ad-bc) x [d -b; -c a]
Inverse (n x n) Row-reduce [A | I] until left side is I -> right side is A-1
Inverse rules (AB)-1 = B-1A-1 | (AT)-1 = (A-1)T | (A-1)-1 = A
A square n x n matrix is invertible if and only if...
Linear Algebra for IEM | Page 2 of 8
Complete Course Summary - University of Groningen - Block 1B
A student-compiled summary covering all midterm and final exam topics: systems of equations, matrix algebra,
determinants, eigenvalues & eigenvectors, vector spaces, geometry, and linear differential equations.
Topics: 1. Systems of Linear Equations | 2. Matrix Algebra | 3. Determinants | 4. Eigenvalues & Eigenvectors |
5. Vector Spaces & Subspaces | 6. Image, Kernel & Rank | 7. Linear Differential Equations | 8. Geometry in
Vector Spaces | 9. Linear Transformations
Linear Algebra for IEM | Page 1 of 8
, 1. Systems of Linear Equations
Compact form: Ax = b, where A is the coefficient matrix, x is the unknown vector, and b is the right-hand side.
No solution Unique solution Infinitely many
RREF gives a false row like 0 = 3. rank(A) = n. Every variable is a At least one free variable (column
pivot variable. without a pivot).
Row Operations
* Eij(d): add d x (row j) to row i
* Ei(c): multiply row i by scalar c (c not 0)
* Eij: swap rows i and j
Gaussian elimination -> RRF (zeros below pivots). Gauss-Jordan -> RREF (leading 1s, zeros above AND below
pivots).
LU Factorisation
A = LU
L = unit lower-triangular (1s on diagonal, Gauss multipliers below).
U = upper-triangular result after forward elimination.
det(A) = det(L) x det(U) = 1 x (product of U diagonal entries).
How to build L: start with the identity matrix and insert the multiplier mij (with original sign) at position (i, j).
Rank and Consistency
rank(A) = number of pivot columns = dim(col space) = dim(row space).
* rank(A) = n -> unique solution (no free variables).
* rank(A) < n -> infinitely many solutions.
* rank([A|b]) > rank(A) -> inconsistent, no solution.
2. Matrix Algebra
Matrix product AB: defined only when cols(A) = rows(B). Result is m x p when A is m x n and B is n x p.
NOTE: AB != BA in general. AB = AC does NOT imply B = C.
Key Identities
Transpose rules (AT)T = A | (AB)T = BTAT | (A+B)T = AT+BT
Inverse (2x2) A = [a b; c d] -> A-1 = 1/(ad-bc) x [d -b; -c a]
Inverse (n x n) Row-reduce [A | I] until left side is I -> right side is A-1
Inverse rules (AB)-1 = B-1A-1 | (AT)-1 = (A-1)T | (A-1)-1 = A
A square n x n matrix is invertible if and only if...
Linear Algebra for IEM | Page 2 of 8
