Math 3000 - Linear Algebra Exam 1

Linearity Properties

Additivity: T(u+v) = T(u) + T(v) for all vectors u, v.

Homogeneity: T(cu) = cT(u) for all scalars c and vectors u.

Parallel Vectors

Two vectors u and v are parallel if one is a scalar multiple of the other, ie: u = kv for some scalar k.

Orthogonal Vectors

Two vectors u and v are orthogonal if their dot product is zero, i.e., u ⋅ v = 0.

Linear Combination

A vector w is a linear combination of vectors v1​, v2​, …, vk​ if w = c1​v1 ​+ c2​v2​ + ⋯ + ckvk​ for some scalars c1​, c2​, …, ck​.

Example: w = (7,7) is a linear combination of v1​ = (1,0) and v2​ = (0,1) because w = 7v1​ + 7v2​.

Span

The span of a set of vectors is the set of all possible linear combinations of those vectors.

Cartesian Equation

Line: ax + by = c

Plane: ax + by + cz = d

Parametric Equation

Line: r(t) = r₀ + tv
r(t) = (1,2) = t(3,4)

Plane: r₀ + su + tv
r(s,t) = (1,2,3) + s(1,0,0) + t(0,1,0)

proj_vu

((u ⋅ v) / (v ⋅ v​))v

What does it mean for a system of equations to be consistent?

A system is consistent if it has at least one solution.

What does it mean for a system of equations to be inconsistent?

A system is inconsistent if it has no solution.

Partial Fraction: 1/(x+4)(x-2)

A/(x+4) + B/(x-2)

Partial Fraction: 1/(x+1)²

A/(x+1) + B/(x+1)²

Partial Fraction: 1/x(x²+3)

A/x + (Bx + C)/(x²+3)

T(u,v)

A[u,v] = [b1,b2]