This is a draft cheat sheet. It is a work in progress and is not finished yet.
VECTORS
2 vectors equal <-> same magnitude + same direction |
norm/magnitude/length of vector ||v|| = sqrt(v_12 +...+ v_n2) |
unit vector -> ||v|| = 1. (v / ||v||) |
||cv|| = |c| ||v|| |
distance d(u,v) = ||u-v|| |
Dot product u•v = (u_1v_1 + ... + u_nv_n) |
n cos(theta) = u•v / (||u|| ||v||) |
u&v are orthagonal when dot(u,v) = u•v = 0 |
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Equations of Lines/Planes
vector form |
x = x_0 + tv |
parametric in R2 |
x = x_0 + ta, y = y_0 + tb |
parametric in R3 |
x = x_0 + ta, y = y_0 + tb, z = z_0 +tc |
General form of plane R3 |
Ax + By + Cz = D |
point normal eq of a plane |
n • (x - x_0) |
vector eq of a plane |
x = x_0 + sv_1 + tv_2 |
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Inverse Matrices
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Inversion Algorithm [ A | I ] -> ... [ | ] ... -> [ I | A-1 ]
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Invertible Matrix Theorem
Invertible Matrix Theorem
If A is an n x n matrix, and if T_A is the linear operator on Rn with standard matrix A, then the following statements are equivalent.
1. A is invertible.
2. The RREF of A is Identity matrix.
3. A can be written as a product of elementary matrices.
4. Ax = 0 has only the trivial solution: x = 0
5. Ax = b has exactly one solution for every vector b in Rn: x = A-1b.
6. Ax = b is consisten for every vector b in Rn.
7. The column vectors of A are linearly independent.
8. The row vectors of A are linearly independent.
9. det(a) ≠ 0.
10. λ = 0 is not an eigenvalue of A.
11. T_A is one-to-one.
12. T_A is onto.
13. The column vectors of A span Rn.
14. The row vectors of A span Rn.
15.The column vectors of A form a basis for Rn.
16.The row vectors of A form a basis for Rn.
17. rank(A) = n
18. nullity(A) = 0
19. A has full column rank. |
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EigenSTUFF
The scalar lambda(λ) is called an Eigenvalue of A when there is a nonzero vector x such that Ax = λx. |
Vector x is an Eigenvector of A corresponding to λ. |
The set of all eigenvectors with the zero vector is a subspace of Rn called the eigenspace of λ. |
1. Find Eigenvalues: det(λI - A) = 0 |
2. Find Eigenvectors: (λI - A)x = 0 |
If A is a triangular matrix then its eigenvalues are on its main diagonal |
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