Find the dimensions of the following linear spaces. (a) The space of all 3×4 matrices (b)...

Find the dimensions of the following linear spaces.
(a) The space of all 3×4 matrices

(b) The space of all upper triangular 5×5 matrices

(c) The space of all diagonal 6×6 matrices

Expert Solution

Colby Messinger answered 2 years ago

For the following matrices, first find a basis for the column space of the matrix. Then...

For the following matrices, first find a basis for the column space of the matrix. Then use the Gram-Schmidt process to find an orthogonal basis for the column space. Finally, scale the vectors of the orthogonal basis to find an orthonormal basis for the column space. (a) [1 1 1, 1 0 2, 3 1 0, 0 0 4 ] b) [?1 6 6, 3 ?8 3, 1 ?2 6, 1 ?4 ?3 ]

Derive the Dirac gamma matrices for 2-space dimensions and 1-time dimension.

Using dynamic programming, find an optimal parenthesization of a matrix-chain product of 4 matrices whose dimensions...

Using dynamic programming, find an optimal parenthesization of a matrix-chain product of 4 matrices whose dimensions are p = { 1, 10, 5, 20, 2}. Show your work.

Find the solution to the heat equation in 3 dimensions on the half space z>0, with...

Find the solution to the heat equation in 3 dimensions on the half space z>0, with homogeneous Neumann boundary conditions at z=0.

1. Find all eigenvalues of each of the following matrices. (a) |10 −18 b) |2 −1...

1. Find all eigenvalues of each of the following matrices. (a) |10 −18 b) |2 −1 (c) |5 4 2 6 −11 | 5 -2| 4 5 2 2 2 2|

For each of the following matrices, find a minimal spanning set for its Column space, Row...

For each of the following matrices, find a minimal spanning set for its Column space, Row space,and Nullspace. Use Octave Online to get matrix A into RREF. A = [4 6 10 7 2; 11 4 15 6 1; 3 −9 −6 5 10]

Let A and b be the matrices A = 1 2 4 17 3 6 −12...

Let A and b be the matrices A = 1 2 4 17 3 6 −12 3 2 3 −3 2 0 2 −2 6 and b = (17, 3, 3, 4) . (a) Explain why A does not have an LU factorization. (b) Use partial pivoting and find the permutation matrix P as well as the LU factors such that PA = LU. (c) Use the information in P, L, and U to solve Ax = b

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Find the dimensions of the following linear spaces. (a) The space of all 3×4 matrices (b)...

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