Linear Algebra ACTUAL EXAM AND PRACTICE
EXAM COMPLETE QUESTIONS AND CORRECT
DETAILED ANSWERS (VERIFIED ANSWERS)
|ALREADY GRADED A+
Cramer's Rule
Ai is the matrix a with column i is replaced by b, where the
problem we're solving is Ax = b.
|det (3x3 matrix)| is geometrically related to
the volume of a parallelepiped whose edges are column
vectors of the matrix (starting at origin)
|det (2x2 matrix)| is geometrically related to
the area of a parallelogram whose edges are column
vectors of a matrix (starting at origin)
Adjugate of a Matrix
the transpose of the cofactor matrix
Eigenvectors
Vectors that remain in the same direction after a linear
transformation. The 0 vector is not allowed.
For matrix A, a vector x
If Ax=λx for a scalar λ, x is an eigenvector
Eigenvalue
,For matrix A, a nonzero vector x
If Ax=λx for a scalar λ, λ is an eigenvalue
Condition for Finding Eigenvalues of a Matrix
Note: Should take out the 0 vector in the set in the last
line.
Trace Of a Matrix
For a square matrix, the sum of its diagonal elements
Sum of Eigenvalues of matrix A =
trace (A)
Product of Eigenvalues (including multiplicities) of matrix A
det (A)
A is sure to have n independent eigenvectors if
all the lamda's are different
If eigen values are repeated then it _________ have n
independent vectors
may or may not
A singular matrix must have an eigenvalue of
0
e^(At) meaning, where a is an nxn matrix and t is a real
number
Matrix Diagonalization
Given an nxn matrix, A with n linearly independent
eigenvalues
A = SΛS^-1
,Where S is the matrix where the ith column is the ith
eigenvalue, and Λ is a matrix where the ith eigenvalue is
on the ith diagonal entry and there are zeros everywhere
else.
Useful for computing powers of many matrices.
General Solution of First Order Difference Equation with
Constant Coefficients
Strang Only
xk represents the kth eigenvector (corresponds to the kth
eigenvalue)
Markov Matrix
Strang Only
All entries of the matrix >= 0
All columns add to one
If a matrix is a Markov matrix, what can you guarantee
about the eigenvalues
Strang Only
1 is an eigenvalue
Complex Eigenvalues of Real Entry Matrices Come in
complex conjugate pairs
Complex Eigenvectors of Real Entry Matrices Come in
complex conjugate pairs
Symmetric Matrix
, A^T = A
2 Fundamental Properties of Real Symmetric Matrices
Eigenvalues are real and eigenvectors can be chosen
perpendicular (If all eigenvectors are distinct, can just say
are perpendicular)
If the eigenvalues of a matrix are real then the number of
positive pivots =
the number of positive eigenvalues
Spectral Theorem
If A is a real symmetric matrix with n independent
eigenvectors, then A can be factored into form A =
QΛQ^−1 = QΛQ^T, where Q is a matrix with n orthonormal
eigenvectors as columns
Positive Definite Matrix
a symmetric matrix with all positive eigenvalues
Positive Semidefinite Matrix
a symmetric matrix with all non-negative eigenvalues
^H operator
Operator that means conjugate transpose
Hermitian Dot Product
Same as standard dot product except the complex
conjugate of the left vector is taken before multiplying
elements
Hermitian Matrices
A^H = A
Given a real symmetric nxn matrix, 4 tests for positive
definiteness are
Eigenvalue test: Check that all eigenvalues are positive
Determinants test: All upper left determinants > 0
Pivot Test: All pivots positive
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