Cheatography
https://cheatography.com
For beginner matrix algebra properties for linear algebra.
This is a draft cheat sheet. It is a work in progress and is not finished yet.
Definition
Leading Variables: variables corresponding to leading 1s. |
Free Variables: variables with no pivot 1s. |
Parameters: symbol (s, t, ...) for free variable. |
Basic Solutions: vectors where s=1 and t,u,..=0. |
Rank A: number of leading 1s. |
If A(m x n) and rank A = r, there are n-r free variables. |
REF: leading 1s with 0s at the bottom. |
RREF: leading 1s with 0s at the top and bottom. |
Main Diagonal: a11, a22, a33, ... entries of a matrix. |
Diagonal Matrix: sq. matrix with entries outside main diagonal = 0. |
Symmetric Matrix: sq. matrix where A = AT. |
Skew-symmetric: square matrix where AT = -A. |
Identity Matrix (I): square matrix where main diagonal entries = 1. |
Zero Matrix: square matrix where all entries = 0. |
Invertible Matrix: sq. matrix that has an inverse. |
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Elementary Row Operations
I. Interchange two rows. |
II. Multiply a row by non-zero integer. |
III. Add multiple of a row with another row. |
Transposition Properties
If A = [aij], then AT = [aji] |
If A is (m×n), then AT is (n×m). |
(AT)T = A |
(kA)T = kAT |
(A+B)T = AT+bT |
Matrix Inverse Properties
AA-1 = I = A-1A |
detA = ad - bc |
adjA = [d -b]
[-c a] |
A is non-invertible if detA = 0. |
A-1 = (1/detA)adjA |
Ax = b ⇒ x = A-1b |
[A I] → [I A-1] |
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Addition and Scalar Multiplication Properties
A+B = B+A |
commutativity |
A+(B+C) = (A+B)+C |
associativity |
0+A = A |
additive identity element |
A+(−A) = 0 |
additive inverse |
k(A+B) = kA+kB |
scalar distributivity |
(k+ p)A = kA+ pA |
scalar distributivity |
(kp)A = k(pA) |
scalar associativity |
1A = A |
multiplicative scalar identity |
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Matrix-Vector Multiplication Properties
x is the solution to Ax = b |
A(x+y) = Ax+Ay |
A(ax) = a(Ax) = (aA)x |
(A+B)x = Ax+Bx |
Associated homogeneous system: Ax = 0 |
Matrix-Matrix Multiplication Properties
IA = A and AI = A |
A(BC) = (AB)C |
A(B+C) = AB+AC |
(B+C)A = BA+CA |
a(AB) = (aA)B = A(aB) |
(AB)T = BTAT |
Idempotent: A2 = A |
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