Linear Algebra Flashcards
Equation to find eigenvalues
det(A-lambda*I)=0
How to find eigenvectors
Solve for the vector (v) that satisfies the equation (A-lambdaI)v = 0.
What does a determinant of zero indicate?
That the matrix does not have linear independence. At least one of the rows is a multiple of another row in the same matrix.
When forming the elimination matrix, how is e_ij formed?
eij = -Aij/Ajj
While forming a 3x3 elimination matrix, after obtaining E32, E21, and E32 separately, what is the shortcut to form E32E321?
a=E21
b=E31
c=E32
E32E321 will have “a” in the 2,1 spot, “c” in the 3,2 spot, and the 3,1 spot will be (a x c) + b
Matrix elimination equation for a 3x3 matrix
E32E321A = E32E321b
NOTE: The key here is that E32 CANNOT be obtained without first finding Abar = E321*A. Once you get E32, you can form the E matrix above and multiply it by the original A.
When forming an elimination matrix of a matrix A, what should be done with matrix A once a new column in the elimination matrix has been completed?
Multiply what you have so far in the elimination matrix by Matrix A. You CANNOT begin to form a new column of E until this step is complete, or the answer will be WRONG.
What is the elimination matrix of A equivalent to?
A^-1
When forming the total elimination matrix E from matrix A, in what order should the columns of E be formed.
Start by forming the lower triangle. Begin with E21 and work to the bottom. Then multiply E and A and move to the next column. Next form the upper triangular by beginning with E12. Multiply E12 and the latest form of A together and then move on to the next column. Finally, finish by forming Ef to reduce all diagonal values of A to one.
Equation for the complete 3x3 elimination matrix of A
Ef E123 E12 E32 E321 A
How do you find the inverse of the elimination matrix E?
E^-1 is made up of the inverse of each of the sub elimination matrices, all multiplied together To invert a sub elimination matrix, simply swap the sign corresponding to the appropriate element.
PA = LU
1) Apply the P matrix at the beginning of the process to get pivots in the correct spots.
2) U is simply the upper triangular matrix formed by using the lower elimination matrix on A.
3) Form L by taking the inverse of E. If E is Ef E123 E12 E32 E321, then L would be E321^-1 E32^-1 E12^-1 E123^-1.
Change the E on the inverted terms to an L, e.g., E321^-1 = L321.
Finally, tack on an Lf term at the beginning of L. Lf is the diagonal of A.
The final result looks like this: A = LPU or A = L321 L32 L12 L123 Lf P U
What does the space of R^n consist of?
All column vectors v with n components
What is a subspace?
A set of vectors (ALWAYS including zero) that satisfies two requirements. If v and w are vectors in the subspace and c is any scalar, then:
1) v + w is in the subspace
2) cv is in the subspace
Basically, all linear combinations of vectors in the subspace with still be in the subspace.
How to find the span of a matrix A?
To find the span of b, break the columns of A up into vectors and multiply each one by x1, x2, x3, etc. Then, pick two sets of values for each of those variables. This will form two vectors. Then find a plane that each of those vectors lies in. Cross the vectors to get the normal vector to the plane. Use a(x-xo) + b(y-yo) + c(z-zo) = 0 to find the equation of the plane/subspace.
Define column space
The column space includes all linear combinations of the columns. The combinations are all possible vectors Ax.
Can Ax=b be solved if b is not in the column space of A?
No
Define N(A), the nullspace of A
N(A) consists of all solutions to Ax=0. The vectors x are in R^n
What are free variables
Free variables are the variables that will be multiplied to the free or dependent columns when trying to find N(A).
What are free columns
Columns that depend on all of the independent columns before them. They come after columns that have pivot values of 1.
What are the steps to find N(A)?
1) Reduce A to row echelon form
2) Locate the independent and the free columns, as well as the free variables
3) Form the S vectors that will make up the null space. There will be the same amount of S vectors as free columns. S vectors are determine by picking numbers for the free variables and then solving for the values of the independent variables.
Full column rank
The columns of A are independent when the rank is r=n
Is a set of vectors dependent or independent when n > m?
Dependent
How to find the rank of a matrix
The rank is equal to the number of independent columns. This is also called the “true column space.” It is also the number of pivots
When are the columns of A linearly independent?
When the only solution to Ax = 0 is x=0. (Vectors cannot simply be combined to give the solution x=0)
Define row space
The row space of a matrix is the subspace of R^n spanned by the rows. It is also the column space of A^T
What is a basis
A basis for a vector space is a sequence of vectors that:
1) are linearly independent
2) span the space
What is the dimension of C(A) and C(A^T)?
r (rank of the matrix)
Dimension of N(A)
n-r
Dimension of N(A^T)
m-r
Define C(A)
The column space of A is the subspace of R^m spanned by the columns
N(A) is a subspace of…
R^n
N(A^T) is a subspace of…
R^m
