Linear Algebra Exercises-n-Answers.pdf

Notation
R real numbers
N natural numbers: {0, 1, 2, . . .}
C complex numbers
{. . . ∣ . . .} set of . . . such that . . .
〈. . .〉 sequence; like a set but order matters
V, W, U vector spaces
⃗v, ⃗w vectors
⃗0, ⃗0 V zero vector, zero vector of V
B, D bases
E n = 〈⃗e 1 , . . . , ⃗e n 〉 standard basis for R n
⃗β, ⃗ δ basis vectors
Rep B (⃗v) matrix representing the vector
P n set of n-th degree polynomials
M n×m set of n×m matrices
[S] span of the set S
M ⊕ N direct sum of subspaces
V ∼ = W isomorphic spaces
h, g homomorphisms, linear maps
H, G matrices
t, s transformations; maps from a space to itself
T, S square matrices
Rep B,D (h) matrix representing the map h
h i,j matrix entry from row i, column j
|T | determinant of the matrix T
R(h), N (h) rangespace and nullspace of the map h
R ∞ (h), N ∞ (h) generalized rangespace and nullspace
Lower case Greek alphabet
name character name character name character
alpha α iota ι rho ρ
beta β kappa κ sigma σ
gamma γ lambda λ tau τ
delta δ mu µ upsilon υ
epsilon ɛ nu ν phi φ
zeta ζ xi ξ chi χ
eta η omicron o psi ψ
theta θ pi π omega ω
Cover. This is Cramer’s Rule for the system x + 2y = 6, 3x + y = 8. The size of the first box is the determinant
shown (the absolute value of the size is the area). The size of the second box is x times that, and equals the size
of the final box. Hence, x is the final determinant divided by the first determinant.