Linear Algebra Formula Sheet
Linear equation Linear Transformations
a1x1+a2+x2+…+an+xn = b T: IRn arrow IRm
Linear Combination T(x) = Ax = x1a1+…+xnan
y=c1v1+c2v2+…+cpvp T(u+v) = T(u) + T(v)
Vector Equations T(cu) = cT(u)
x1a1+x2a2+…xpap = b Standard Matrix Theorem
[a1 a2 …. ap | b] A = [ T (e1) T(e2) … T(en)]
Span {v1, v2, …vp} Standard Basis Vectors
Matrix Equation e1 = [1 0 0], e2 = [0 1 0], e3 = [0 0 1]
Ax = b Powers of a Matrix
Identity Matrix Ak, n x n matrix
n x n matrix A0 = In
Inx = x Transpose Properties
Matrix – Vector Properties (AT)T = A (AB)T = BTAT
A(u+v) = Au + Av (A+B)T = AT + BT (rA)T = rAT
A(cu) = c(Au) Matrix Inverses Properties
Parametric Vector Form A-1 A = I AA-1 = I
x = tv, t span of IR (AB)-1 = B-1A-1 (A-1)-1 = A
x = su + tv, s, t span of IR A-1 = 1/ad-bc [d, -b, -c, a] x = A-1b
[A | I ] ~ [ I | A-1 ]
Linear transformations properties Matrix Multiplication Properties
T (0) = 0 A(BC) = (AB) C ImA = A = AIn
T(cu+dv) = cT(u) + dT(v) A(B+C) = AB + AC
T (c1v1+…. +cpvp) = c1T(v1) +…+ cpT(vp) r(AB) = (rA)B = A(rB)
Linear equation Linear Transformations
a1x1+a2+x2+…+an+xn = b T: IRn arrow IRm
Linear Combination T(x) = Ax = x1a1+…+xnan
y=c1v1+c2v2+…+cpvp T(u+v) = T(u) + T(v)
Vector Equations T(cu) = cT(u)
x1a1+x2a2+…xpap = b Standard Matrix Theorem
[a1 a2 …. ap | b] A = [ T (e1) T(e2) … T(en)]
Span {v1, v2, …vp} Standard Basis Vectors
Matrix Equation e1 = [1 0 0], e2 = [0 1 0], e3 = [0 0 1]
Ax = b Powers of a Matrix
Identity Matrix Ak, n x n matrix
n x n matrix A0 = In
Inx = x Transpose Properties
Matrix – Vector Properties (AT)T = A (AB)T = BTAT
A(u+v) = Au + Av (A+B)T = AT + BT (rA)T = rAT
A(cu) = c(Au) Matrix Inverses Properties
Parametric Vector Form A-1 A = I AA-1 = I
x = tv, t span of IR (AB)-1 = B-1A-1 (A-1)-1 = A
x = su + tv, s, t span of IR A-1 = 1/ad-bc [d, -b, -c, a] x = A-1b
[A | I ] ~ [ I | A-1 ]
Linear transformations properties Matrix Multiplication Properties
T (0) = 0 A(BC) = (AB) C ImA = A = AIn
T(cu+dv) = cT(u) + dT(v) A(B+C) = AB + AC
T (c1v1+…. +cpvp) = c1T(v1) +…+ cpT(vp) r(AB) = (rA)B = A(rB)
