MIT 18.06 Linear Algebra course
Lecture 1
We learn about the big picture behind multiplication of matrix and vector
we learn about the row picture and column picture
Lecture 2
we learned about elimination method to solve a system of equations
Lecture 3
in this lecture we learned about matrices multiplication:
we can do that in five ways:
- row * col ==> gives an entry (1 cell)
- col _ row ==> sum ( r1 _ c1 , r2 * c2, etc)
- by columns ==> A * c1 = combination of A columns
- by columns ==> r1 * B = combination of A B rows
- by blocks ==> A (A1,A2,A3,A4) _ B (B1,B2,B3,B4) = C1 = (A1_ B1 + A2 * B3) and so on
then we learned about gausian-Jordan elimination to find the matrix inverse
[A | I] ==> we apply elimination till we get [ I | A-1 ]
Lecture 4
in this lecture we learn about A= L U, where L is E^ -1, and whats special about this is that it has all multipliers in the lower triangular with ones on the diagonal
Lecture 5
we continued a little with permutations and moved to vector spaces we learned about sub spaces and columns spaces ==> which is u take the columns of the matrix and all its combinations and that creates a plane through origin making a columns space
Lecture 6
In this lecture we continued about columns spaces and that we build those up by taking the combinations of all columns. Then we learned about null spaces while are sub spaces of X that satisfies A X = 0
Lecture 7
in this lecture we continued about null space
then we learned about the special solution, where we assume the variables of the free vector then get the special solution
finally we learned about the reduced form
where R = [ I F 0 0 ]
and the null matrix is [ -F I ]
then R N = 0
Lecture 8
in this lecture we expanded to talk about A x = b
and we find the whether there’s a solution to the equation or not depends on the rank of the matrix
also we get the Xcomplete = Xparticular + Xnull space and we get particular soln by putting all free variables = zero
Lecture 9
in this lecture we learned about independent columns and how they make a space, we also learned about Basis and what are two conditions for it
rank(A) = number of pivot columns of A = dimension of C(A).
dimension of N(A) = number of free variables = n − r,
Lecture 10
In this lecture we learned about the four subspaces
we also started in matrix space M
Lecture 11
We learned about matrix space we take introduction about graph
Lecture 12
in this lecture we learned about graphs and how to represent them with matrices, then we applied that to electrical system and applied kerchofs law
Lecture 13
Quiz 1 review
Lecture 14
in this lecture learned about othrignilaity of the four vector spaces and what does it means
Lecture 15
in this lecture we learned about projection of matrices into subspaces
Lecture 16
we got example explaning the projection into subspaces and how to get the best fit using the least square error
Lecture 17
in this lecture le learned about orthonormal vectors and their special features and we learn how to produce them from any independent vectors using gram-schmeit
Lecture 18
Propertise of determants
Lecture 19
- det I =1
- sign reverses with each row or colums exchange
- det is linear in each row seperately
Big Det Formula
for a N * N matrix, we calc the sum of N! terms
$$detA=\sum_{i=1}^ N a1\alphaa2\betaa3\gamma*an\omega$$
(where $\alpha, \beta, … \omega $) = perm of (1,2,3, …, N )
Cofactors
cofactor of aij = Cij = +/- det of ( n-1 matrix with column j, and row i erased )
it is plus if i+j is even, minus if i+j is odd
cofactor formula (along row 1)
det A = a11 C11 + a12 C12 + .... + a1n C1n
Lecture 20
$ A^{-1} = 1/detA * C^T$ where C is the cofactors matrix
Cramers rule
A x = b
x= A^ (-1) b = 1/detA C^T b
X_j = detB_j / detA where B_j is A matrix with column j replaced by b
Det A = Volume
detA = volume of the shape created by making an edge from each of the rows
Lecture 21 Eigenvalues and Eigenvectors
Eigenvectors: Ax is prallel to x ==> Ax = $\lambda$x
lambda ia the eigen values
if we have a plane:
- any x in the plane: Px= x ==> x is eigenvector and lambda = 1
- any x perpendicular to plane Px = 0 ==> x is eigen vector and lambda = 0
Fact: the sum of the eigenvalues = the sum of the diagonal of A
Lecture 22: Diagnolization
to get power of matrix $A^k$
first get the eigenvalues and vectors for A
then compute $A = S \lambda S^{-1}$ where S is the eigenvector matrix, and Lambda is diagonal matrix of the eigenvalues
then $A^k = S * \lambda^k*S^{-1}$
Lecture 23
for the diffrential equations:
1- Stability if lambda < 0 ==> u(t) –> 0
2- Steady state if lambda1 = 0, lambda2 < 0
3- Blowup if any lambda > 0
Lecture 24
Markov Matrix
1- All entries >= 0 2- The sum of every column is 1
3- lambda = 1 is eigenvalue
4- all othe lambda < 1 5- eigenvector values >= 0
Fourier series
integration of ( f(x) g(x) dx ) from 0 to 2pi = 0
Lecture 25
Symmetric matrices
$ A = A^T$
- the eigenvalues are real
- the eigenvectors are perpendicular
usual case:
$A = S \lambda S^{-1}$
symmetric case:
- we have orthonormal eigenvectors
$A = Q \lambda Q^{-1} = Q \lambda Q^{-T} $
Every symmetric matrix is a combination of perp. projection matrices
Signs of pivots are the same as the sign of the eigenvalues
product of pivots = product of eigenvalues = det of matrix
Positive definite symmetric matrix
all eigenvalues are positives
all pivots are positive
det is positive as it’s the product of the eigenvalues
also all sub detemants are positive ( determants of lower matrices )
if S is pos. definite ==> $ X^TSX$ must be positive
Lecture 26
Complex Matrices
we wanna utilize tha fact that $\bar{Z^T}*Z = \left|{Z}\right|^2$
Hermitian is biscally the conj. and transpose ==> $Z^H = \bar{Z^T}$
Hermitian Matricies : $A^H = A$
perpendicular: q1, q2, …, qn
- $\bar{qi}^T * qj = 0 if i!=j, 1 if i=j $
- $Q^H*Q = I$
Fourier Matrix
a matrix with entries are powers of some number W. where $W^n = 1$
$ F^H*F = I$
Fast fourier transform
- reduces complexity from $N^2 to N log(N)$
- $W_{2n}^2 = W_n ==> W_4^2 = W_2$
Lecture 27
when det= 0 ==> then the matrix is positive semi-definite
f(x1,x2,x3…,xn) ==> min when the matrix of second derivatives is positive definite
- the eigenvalues tells us the length of the axis of the shape crated by cutting through the shape of the $X^T A X $
- the direction of the eigenvectors is the direction of the axis of that shape
Lecture 28
A is a m by n matrix ==> $ A^T*A$ is a positive definite symmetric matrix
similar matrices
- A and B are similar means: for some M ==> $B = M^{-1}AM$
- Similar matrices has the same eigenvalues if the eigenvalues are unique
- the eigenvector of B is $M^{-1} * (eigenvector ofA)$
Jordan form
- every square A is similar to Jordan matrix J
- every jordan block has one eigenvaector
- the number of jordan blocks = number of eigenvectors
Lecture 29
- eigenvalues of (AB) = eigenvalues of (BA)
Singular value composition (SVD)
- $Av = \sigma u $
- $ A = u \sigma v^T = u \sigma v^{-1}$
- $A^T A = v \sigma^T u^Tu /sigma v^T = v \sigma^2 v^T$
- $A A^T= u \sigma^T v^Tv /sigma u^T = u \sigma^2 u^T$
Lecture 30
Linear transformation
examples: projection, rotation
if u know what transfotmation does to the basis of a plane, then u know what it does to every vector in the plane
- every $ v = c_1 v_1 + c_2 v_2 + … + c_n v_n$
- then $T(v) = c_1 T(v_1) + …. + c_n T(v_n)$
coordinates come from a basis (think of basis like the X-Y-Z axis
so we have a basis for the input and a basis for the output
Rule to find matrix A, given the input and output basis :
- input basis: v1 ===> vn
- output basis: w1 ===> wm
- 1st column of A : write T(v1) = a11 w1 + a21 w2 + … + am1 wm
- 2nd column of A : write T(v2) = a12 w1 + a22 w2 + … + am2 wm
- repeat that for the n columns
A * (input coordinates) = (output coordinates)
Lecture 31
change of basis
- we have a new basis vectors and we wanna change to the new basis W
- A = c _ W ==> c = W^-1 _ A
- when we change the basis, every vector would have new coordinates ==> old coordinates = new basis * new coordinates ==> x = W c
Lecture 32
2-sided inverse
- $A A^{-1} = I = A^{-1} A$
- r = m = n ==> full rank
left inverse
- full column rank
- r = n
- nullspace = 0
- then $A^T A $ is invertable
- $A^{-1}_{left} = (A^T A)^{-1} A^T $
- $A^{-1}_{left} A = I$
right inverse
- full row rank
- r = m
- n-m free variables
- left nullspace = 0
pseudo inverse $A^+$
$A^+ = v*\sigma^{-1}*u^T$
Lecute 34
- no solution ==> rank < m
- has one solution ==> there’e no null space ==> rank = n
- a matrix is invertable when there’s no null space ==> r = n ==> indep. columns
- positive definite matrix must have full rank ==> has no null space
- positive def is invertable
- the matrix has soln of any c when the matrix has full row rank
- matix with orthogonal eigen vectors :
- symmetric matrices
- skew-symmetric
- orthogonal matrices
- in markov matrix
- the eigen values are one, and some sother values less than one
- $k_m$ and m goes to infinity is the steady state we ge the eigenvector that corresponts to eigenvalue one asn multiply it with c, and notes that the sum of u is alwayes the same, so the sum of u0 is the sum of uk, so look what c achieves that
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