Karteikarten
Linear Algebra T1
Karten in diesem Stapel (34)
- 1what's a set?Paper Bag => stuff in it => if it doesn't, then it's an empty set.
- 2What's span?Set of solutions
linear combination - 3when we have: -System of equation -Vector equation -Matrix equationwe always start by setting up the matrix
- 4Is the vector {w} in the span of {v1,v2,v3}>x[v1] + y[v2] + z[v3] = W
then we set the matrix and solve - 5Let A be a matrix, describe all the solutions for Ax = 0Set as a matrix and solve the matrix, solve for a variable (x,y or z) then set it. (imagine we solve for x)
[x] = [what we found solving for x]
[y] = [ y]
[z] = [ z]
[w] = [ w]
then we factor our x,y,z and w - 6Linear combinationsexamples:
- 7matrix * vectorthe matrix is always on the left
[a*b]
a= rows
b=columns
the # of entries in the vector must be the same as the # of columns in the matrix
inside number tell if the multiplication is possible or not - 8Reduce row echelon form (make pivots numbers = 1)a
- 9Matrix * matrixthe # of rows in the first matrix must be the same # of columns in the second one (or opposite0
result is the product of the left side of the matrix time the right side of the second matrix
result => [a*b]*[c*d] = [a*d]
*matrix multiplication is not commutative* - 10transpose of a matrixrows become columns, columns become rows
represented by a T in the up right corner - 11linear independencevectors are linearly independent if and only if there is only one solution to the equation.
only trivial solution = 0 solution, there are non-trivial solutions. - 12hints for linear independence- 2 vectors multiple of each other => not linearly independent
- If you have more vectors than entries per vector, then not linear independent.
-if one set of vectors is zero, then not independent - 13examples of linear independency
- 14theorem about dependency relationIf vectors are non-independent, at least one is a linear combination of the others. (dependency relation)
- 15linear transformationif T is a function with domain R^m and codomaind R^n, then T is a linear transformation if and only if:
1) T(x+y) = T(x) + T (y)
2) T(cx) = cT(x) - 16how to show T is a linear transs ?just extract the matrix
- 17how to show T is not a linear transformation?pick a random scalar and vector, prove they don't match
- 18Find all "a" When A=0Work with determinant till you reach a polynomial equation.
- 19singularityWhen the determinant is 0.
- 20Matrix is singularDeterminant is equal to 0
- 21Vectors independenceA relationship between them = dependent
set up a dependence equation and if the equation has trivial solution , then LI - 22Giving a dependence relation (equation)Use the general equation(C1V1+C2V2+...+CnVn=0)
plug in values found in terms of the dependent vector - 23inverse of a 2x2 matrix
- 24Invertible or not invertible?0 for all V (NI)
V for all V (I)
Rotation about x-axis (I)
Dilation by a factor of 8 constant (I)
Reflection into yz plane (I)
Projection y axis (NI) - 25Finding E1 inverse (2x2 matrix)left side times inverse of right side = E1
then take the inverse of E1. - 26Inverse 3x3multiply by identity matrix
- 27inverse of a linear transformationfind the inverse of the 2x2 matrix, then multiply by the vector.
- 28Onto matrixvectors equal to a,b,c
- 29one to onevectors equal to 0
- 30Determinant 2x2a*d - b*c
- 31Determinant 3x3knock out columns and rows
- 32Basisdependent = no basis
- 33Rotationsuse (xcos(angl)+ysin(angle), -xcos(angle)+ysin(angle)
- 34—
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- Beschreibung
- Mathematics Karteikarten am Linear Algebra T1, erstellt von Chimi Ponce am 08/02/2023.
- Erstellt von
- Chimi Ponce
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