(1.) (a) Suppose A,B,C are matrices with the same number of rows. Multiply the following block matrices:
$[ A \ B \ C ]^T[ A \ B \ C ]$ and $[ A \ B \ C ][ A \ B \ C ]^T$ in terms of A,B,C and transposes.
(b) Suppose further that $[A \ B \ C]$ is a (square) orthogonal matrix. What can you say about $A^TA$,$B^TB$,$C^TC$ and $A^TC$,$B^TC$,$A^TB$?
(c) Again assuming $[A \ B \ C]$ is (square) orthogonal, what can you say about $(AA^T)^2$,$(BB^T)^2$,$(CC^T)^2$? The three matrices $AA^T$,$BB^T$,$CC^T$ are called ___________ matrices?
(2) Consider the vector space F of functions of the form $f(x) = a+bx+cx^2 + d \sin(x)+e \cos(x)$.
(2a) What kinds of functions should we say live in the nullspace of d/dx?
(2b) What kinds of functions are in the analog of the column space? (If the column space is {Av} for column vectors, then perhaps it should be {f' for f in F}.)
(2c) What should we call the rank of d/dx acting on F?
(2d) If f(x) is in the "column space", what is the analog of the solution to Ax=b?
(2e) are there 0,1, or infinitely many solutions?
(3 GS p. 175) Find the largest possible number of independent vectors among [1,-1,0,0], [1,0,-1,0],[1,0,0,-1],[0,1,-1,0],[0,1,0,-1],[0,0,1,-1]?
(4) Consider the matrix A below
using LinearAlgebra
A =
[0.138803 0.0635682 0.16645913
0.382691 0.393285 0.59615
0.373926 0.3696142 0.573268
]
3×3 Array{Float64,2}: 0.138803 0.0635682 0.166459 0.382691 0.393285 0.59615 0.373926 0.369614 0.573268
svdvals(A)
3-element Array{Float64,1}: 1.1444252264762649 0.05383825009216254 3.848754866158789e-10
(4a) Explain why a pure mathematician would say, yes, this matrix A has three independent columns.
(4b) Explain why a more practically minded applied mathematician would says, that A nearly has three dependent columns.
You can use Julia or not in your explanations.
(5) Suppose S is a 2-dimensional subspace of $R^3$. Can every basis of S be extended a basis of R^3? Can every basis of R^3 be reduced to a basis for S?
(6) (similar to GS 3.4 24 p.177) True or false (give an SVD based reason)
(6a) If athe columns of a metrix are dependent, so are the rows.
(6b) The column space of a 2x2 matrix is the same as its row space.
(6c) The column space of a 2x2 matrix has the same dimension as its row space.
(6d) The columns of a matrix are always a basis for the column space.
(7) For a fixed n, what is the span of all nxn invertible matrices? All nxn singular matrices? All nxn upper triangular matrices?
(8) (GS 3.4 39 p.179 ) Supose A is 5x4 with rank 4. (8a) Show that Ax=b has no solution when the 5x5 matrix $[A \ b]$ is invertible.
(Hint: think about independence)
(8b) Show that $Ax=b$ is solvable when $[A \ b]$ is singular.
(9) Consider all 3x3 matrices A that satisfy A*[1,1,1] is a multiple of [1,1,1]? Do these form a subspace of 3x3 matrices? (Matrices with all row sums equal to each other)
(10) (GS p.178 problem 35) Find a basis for the space of polynomials p(x) of degree ≤ 3? Find a basis for the subspace with p(1)=0