Exam (elaborations)
CHE505 Final Exam With Complete Solutions Latest Update
- Course
- CHE 505
- Institution
- Arizona State University
CHE505 Final Exam With Complete Solutions Latest Update
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Solution 2024/2025Pepper
CHE505 Final Exam With Complete Solutions
Latest Update
(1) When would we need to solve Ax=b ANS✔✔ Systems of linear eqns,
interested in steady states of processes, fitting exp. data, simulating ODEs
and PDEs, optimization
(1) Cramers Rule ANS✔✔ Find the kth element of vector x from Ax=b. As
long as A is not singular, a solution exists and is unique.
(1) What are the scalability challenges of Cramer's Rule ANS✔✔ Scales with
the size of matrix factorial. Follows combinatorial scaling: o(n!).
(1) Describe back solving a matrix ANS✔✔ Generate a U matrix, then use
previous elements to solve Ax=b.
(1) Describe process of Gaussian Elimination (GE) ANS✔✔ Complete
elementary row operations by multiplying row by scalars then adding rows
together.
(1) Describe an augmented matrix ANS✔✔ Syntax: (A|b): combines A matrix
and b vector into a single matrix, pasted.
(1) Describe the scaling of GE ANS✔✔ GE scales based on the number of
multiplication and addition operations. Follows polynomial scaling: o(n^3)
, Solution 2024/2025
Pepper
(1) What is the determinant of the U matrix ANS✔✔ Product sum of diagonal
elements
(1) Describe pivoting and why it is useful ANS✔✔ Uses row swaps to ensure
we can proceed with GE. Necessary when diagonal term is 0. Useful when
diagonal term is <<1 and nonzero (can lead to round off errors without).
Each pivot changes sign of the determinant.
(1) Describe LU decomposition and why it is useful ANS✔✔ Factor matrix A
into product LU s.t. Ly=b and Ux=y. Store L and U in memory to just
back/forward solve
(1) When can LU decomposition be completed (when does it exist) ANS✔✔
When the principal minor |Ap| nonzero for p=1,...,n-1
(1) What are L and U ANS✔✔ U: matrix that results from GE. L: matrix with
1's on main diagonal; below main diagonal are the negative multipliers from
GE.
(2) When does GE work without pivoting ANS✔✔ When A is diagonally
dominant
(2) Describe positive definite ANS✔✔ For any vector x=!0, (x^T)Ax>0
(2) Give examples of uses for banded matrices ANS✔✔ Finite difference on
diff. eqn., chemical process of units/stages in series
, Solution 2024/2025
Pepper
(2) What is the formula for bandwidth ANS✔✔ p+q+1 (p is nonzero rows and
q is nonzero columns)
(2) Describe LU decomposition's relation with banded matrices ANS✔✔ LU
decomp. will retain banded structure
(2) Describe a linear vectore space ANS✔✔ Collection of vectors, matrices,
and functions: S. Shares certain properties we can dervie such as addition,
scalar multiplication, additive inverse, zero, commutative, distributions,
associative. Defines the algebraic structure to our collection of objects.
(2) Describe the vector norm ANS✔✔ Introduces a notion of size or length of
a vector.
(2) Properties of vector norms ANS✔✔ 1) ||x||>=0, ||x||=0 iff x=0. 2) ||ax|| =
|a|||x||, a is a scalar. 3) ||x+y||<=||x||+||y||
(2) Describe a linear operator ANS✔✔ An operator in vector space transform
x in S to new object y in S.
(2) What makes an operator linear ANS✔✔ Superposition [A(x+y)=Ax+Ay]
and Homogeneity [A(ax)=a(Ax)]
(2) What does the matrix norm tell us ANS✔✔ Maximum amplification of
applying operator to vector
(2) Properties of matrix norms ANS✔✔ 1) ||A||>=0, ||A||=0 iff A=0. 2) ||aA||
=|a|||A||. 3) ||A+B||<=||A||+||B||. 4) ||AB||<=||A||||B||
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