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Linear Algebra
1 a) Show that if A is nonsingular symmetric matrix, then A^-1 is also symmetric. Write your
justi cation in clear sentences.
b) An n x n matrix A is called skew-symmetric if A = -A^T . Show that if n is odd a skew-symmetric matrix is singular.
justi cation in clear sentences.
b) An n x n matrix A is called skew-symmetric if A = -A^T . Show that if n is odd a skew-symmetric matrix is singular.
Linear Algebra
1.a) Prove that the product A = v(w^T) of a nonzero m x 1 column vector v by a nonzero 1 x n
row vector w^T is an m x n matrix of rank 1. [Hint: do a few small examples]
b) Now show that if A is an m x n matrix of rank 1, then there exist a nonzero m x 1 column
vector v and a nonzero 1 x n row vector w^T such that A = v(w^T).
row vector w^T is an m x n matrix of rank 1. [Hint: do a few small examples]
b) Now show that if A is an m x n matrix of rank 1, then there exist a nonzero m x 1 column
vector v and a nonzero 1 x n row vector w^T such that A = v(w^T).
Linear Algebra
Is orthogonality reflexive, symmetric, and transitive? If so, it is an equivalence relation. If not true, find a counter-example.
Linear Algebra
Which is easier to compute, the U in the LU decomposition or the R in the QR decomposition? Explain your reasoning?
Linear Algebra
Rate QR factorization with the LU and PLU factorizations. Which would you prefer when solving large systems of linear equations, and why?
Linear Algebra
Create a one page “cheat sheet” in flow-chart form which explains the Gram-Schmidt process.
Linear Algebra
Suppose u have the value of error sum of square is 9.1,so what would u comments on this value ?
data
X y
4 10
5 14
6 12
7 17
8 19
data
X y
4 10
5 14
6 12
7 17
8 19
Linear Algebra
1. (2 Points) Explain clearly why the solution to the homogeneous system Ax = 0 with a nonsingular
coecient matrix is x = 0.
2. (2 Points) Under what conditions does a diagonal matrix D = diag(d1,d2,.....,dn) have an inverse
D^-1? What is the inverse D^-1 when these conditions are met? Justify your answers.
3. (3 Points) Let A be an mxn matrix and B be an nxm matrix where m > n. Show that the nxn matrix AB is not invertible.
coecient matrix is x = 0.
2. (2 Points) Under what conditions does a diagonal matrix D = diag(d1,d2,.....,dn) have an inverse
D^-1? What is the inverse D^-1 when these conditions are met? Justify your answers.
3. (3 Points) Let A be an mxn matrix and B be an nxm matrix where m > n. Show that the nxn matrix AB is not invertible.
Linear Algebra
1. (2 Points) Explain clearly why the solution to the homogeneous system Ax = 0 with a nonsingular
coecient matrix is x = 0.
2. (2 Points) Under what conditions does a diagonal matrix D = diag(d1,d2,.....,dn) have an inverse
D^-1? What is the inverse D^-1 when these conditions are met? Justify your answers.
3. (3 Points) Let A be an mxn matrix and B be an nxm matrix where m > n. Show that the nxn matrix AB is not invertible.
coecient matrix is x = 0.
2. (2 Points) Under what conditions does a diagonal matrix D = diag(d1,d2,.....,dn) have an inverse
D^-1? What is the inverse D^-1 when these conditions are met? Justify your answers.
3. (3 Points) Let A be an mxn matrix and B be an nxm matrix where m > n. Show that the nxn matrix AB is not invertible.
Linear Algebra
Show that, for any vector z in C^J, the member of H_i closest to z is x having the entries:
___
x_j = z_j + a_i^(-1)A_ij(b_i - (Az)_i),
where a_i = The sum from j=1 to J of |A_ij|^2.
___
x_j = z_j + a_i^(-1)A_ij(b_i - (Az)_i),
where a_i = The sum from j=1 to J of |A_ij|^2.
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#339914 on Dec 2023
#339914 on Dec 2023