linear chapter 5

coordinate vector equation (how to get the LC constants)

c = w dot v_i over v_i²

how to find all possible v given v1 and that v2 is a vector x, y

• use the dot product of v1 and v2 = 0

• use x² + y² = 1

• solve for all cases of x and y (there will be two options)

• solve for what v equals when v2 is one or the other, using coord vector as LC

how to show a matrix is orthogonal

• Q^TQ = I_n

• or, show that each row is orthogonal to the next and magnitudes for each column are 1

how to find the basis of a line

• solve for one variable

• so like x = 5/2 y

• set one variable equal to t (so y = t)

• solve for the span, which is t [5/2, 1]

• multiply to make it [5, 2]

• if finding an orthogonal complement, find the perpendicular line, and create a basis out of that

how to normally find a basis for the orthogonal complement

solve for the null tranpose space

how to find the four funademtnal subspaces

• first, find RREF

• from there, look at pivot columns

  • row A = any non-zero row in RREF

  • col A = the corresponding pivot columns (has leading 1s) of og matrix

  • null A = null space

  • null A^T = null space of the transpose

how to find the orthogonal decomposition

v = w + w^orth

v = perp_wv + proj_wv

proj_wv = the summation of (u_k dot v / u_k²) * u_k

perp is just v minus that

when would you use G/S process and how to use it

• first check if they’re orthogonal: if they are, no need to do G/S

• you use it to find an orthogonal basis for a span where you’re given x1, x2, x3

• v_k = x_k - (v₁ dot x_{k /} v₁²) * v_{1 - etc - etc until you get to k being k - 1}

  • if you’re finding v3, you start with x3 minus everything

  • the only other part you use x3 is for the dot product. everything else will be v1 or v2

• once you find all your vectors, put them in a span together and take out denominators as necessary and that’s the orthogonal basis

how to do QR factorization

R = Q^TA

put all the vectors you found using G/S into a matrix, take its tranpose, and normalize it

that’s Q, and A will be given

how to find an orthogonal basis for any subspace where x1, x2, x3 aren’t neatly given

• first check if they’re orthogonal: if they are, no need to do G/S

• parameterize to find the span

• use G/S process to find vectors and put them in a basis (multiply by constants to make them easier)

• you can make up random vectors like 0 0 0 1 if you need more than you are given

how to orthogonally diagonalize a square matrix (find Q)

• find its eigenvectors

• test if they are orthogonal, if they are not, use G/S (test all combos, if one of them from a dif lambda is orth [] there’s no reason to do G/S again)

• put them all in a matrix together and normalize them (don’t take transpose here)

how to find spectral decomposition

eigenvalue 1 [normalized eigenvector] [normalized eigenvector]^T + eigenvalue 2 [normalized eigenvector] [normalized eigenvector]^T + eigenvalue 3 [normalized eigenvector] [normalized eigenvector]^T

essentially just take the LC of Qcol1 times Qcol1 as a row with lambda as the constant, and repeat for how many cols you have

what does spectral decomp give you, if evaluated

the symmetric matrix A such that Q^TAQ = D

thus, A = QDQ^T

rotation matrix

[cos -sin sin cos]

det = 1

reflection matrix

[cos2theta sin2theta sin2theta -cos2theta]

det = -1

relations between dim and nullity

dimRowA + nullityA = n

dimColA + nullityA T = m

dimRowA = dimColA = rankA

relations between 4 fundamental subspaces

col A is the orthogonal complement of null A T

row A is the orthogonal complement of null A

what are D and P

D is a matrix of eigenvalues

P is the matrix of eigenvectors, so Q before it gets normalized

SHOW BOTH!