Let A be some m*n matrix. Consider the set S = {z : Az = 0}....

Let A be some m*n matrix. Consider the set S = {z : Az = 0}. First show that this is a vector space. Now show that n = p+q where p = rank(A) and q = dim(S). Here is how to do it. Let the vectors x₁, . . . , x_p be such that Ax₁, . . . ,Ax_p form a basis of the column space of A (thus each x can be chosen to be some unit vector with a 1 corresponding to the position of a column vector that is part of a (maximally) linearly independent set) and let the vectors z1, . . . , zq be a basis for S. Then show that the two sets together i.e. the set {x1, . . . , xp, z1, . . . , zq} form a basis of the n-dimensional Euclidean space.

Using the result above, offer a direct proof of the result r(X′X) = r(X) without appealing to the product rank theorem

Expert Solution

milcah answered 1 year ago

Let A be an n × n matrix which is not 0 but A2 = 0....

Let A be an n × n matrix which is not 0 but A2 = 0. Let I be the identity matrix. a)Show that A is not diagonalizable. b)Show that A is not invertible. c)Show that I-A is invertible and find its inverse.

Let A be an m × n matrix and B be an m × p matrix....

Let A be an m × n matrix and B be an m × p matrix. Let C =[A | B] be an m×(n + p) matrix. (a) Show that R(C) = R(A) + R(B), where R(·) denotes the range of a matrix. (b) Show that rank(C) = rank(A) + rank(B)−dim(R(A)∩R(B)).

Consider the relation R defined on the set Z as follows: ∀m, n ∈ Z, (m,...

Consider the relation R defined on the set Z as follows: ∀m, n ∈ Z, (m, n) ∈ R if and only if m + n = 2k for some integer k. For example, (3, 11) is in R because 3 + 11 = 14 = 2(7). (a) Is the relation reflexive? Prove or disprove. (b) Is the relation symmetric? Prove or disprove. (c) Is the relation transitive? Prove or disprove. (d) Is it an equivalence relation? Explain.

(2) Let Z/nZ be the set of n elements {0, 1, 2, . . . ,...

(2) Let Z/nZ be the set of n elements {0, 1, 2, . . . , n ? 1} with addition and multiplication modulo n. (a) Which element of Z/5Z is the additive identity? Which element is the multiplicative identity? For each nonzero element of Z/5Z, write out its multiplicative inverse. (b) Prove that Z/nZ is a field if and only if n is a prime number. [Hint: first work out why it’s not a field when n isn’t prime....

Let f : Z × Z → Z be defined by f(n, m) = n −...

Let f : Z × Z → Z be defined by f(n, m) = n − m a. Is this function one to one? Prove your result. b. Is this function onto Z? Prove your result

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0....

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0. (b) Show that A is symmetric if and only if A^2= AA^T

Let S be a set of n numbers. Let X be the set of all subsets...

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Let the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider...

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Let A be a m × n matrix with entries in R. Recall that the row...

Let A be a m × n matrix with entries in R. Recall that the row rank of A means the dimension of the subspace in RN spanned by the rows of A (viewed as vectors in Rn), and the column rank means that of the subspace in Rm spanned by the columns of A (viewed as vectors in Rm). (a) Prove that n = (column rank of A) + dim S, where the set S is the solution space...

Let A be an m x n matrix and b and x be vectors such that...

Let A be an m x n matrix and b and x be vectors such that Ab=x. a) What vector space is x in? b) What vector space is b in? c) Show that x is a linear combination of the columns of A. d) Let x' be a linear combination of the columns of A. Show that there is a vector b' so that Ab' = x'.

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