Mathematics
Columbia University Linear Algebra and Square Matrices Questions

Columbia University

### Question Description

I need support with this Algebra question so I can learn better.

question 1 has 5 true and false question

detailed description is in the attached file.

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Linear Algebra (MTH-SHU 140) Quiz of week 6 2020/03/27 The Lecturer Name Instructions. Please write down your answers in details. Let R denote the set of real numbers. 1. Is the following each statement true or false? If false, write F and give counter examples. If true, write T and give brief explainations. (1) Let An m be a matrix. Then rank(P A) = rank(A) for any invertible matrix Pn (2) Let u; v be nonzero vectors in Rn : Then rank(uv T ) = n: rank(v T u): (3) Let An m be a matrix and Pn n an invertible matrix. Then column space of A is the same as that of P A; i.e. Spanfcolumns of Ag = Spanfcolumns of P Ag: (4) Let A; B be two matrices of the same sizes. Then dim N ul(A)+dim N ul(B) B): dim N ul(A+ (5) Supose that AB = C for three square matrices A; B; C: If the rows of C are linearly independent, then the rows of B are linearly independent. 1 1 ; u2 = 1 two sets of vectors in R2 : 2. Let U = fu1 = 1 cos g and V = fv1 = 1 sin ; v2 = sin cos g (for any (1) Prove that U; V are bases for R2 : (2) Find the transition matrix P from U to V; i.e. [x]V = P [x]U for any x 2 R2 : 2 2 R) be Bn m Dn 0 Cm rank(B) + rank(C): 3. Let A = l be a partitioned matrix for matrices B; C; D. Prove that rank(A) = l 3 4. Let M2 be the vector space of all 2 2 matrices. De…ne a function T : M2 a b T (A) = A + AT for any A = 2 M2 2 : c d 2 2 ! M2 (1) Show that T is linear. (2) Find a basis of M2 2: (2) Find a standard matrix AT of the linear map T and compute the rank of AT : 4 2 by ...
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Question one
Part 1: T
For any invertible matrix P, there exists the condition that Rank (PA) = Rank (A0 such that row
vectors of PA are a linear combination of the rows of A.
Part 2: T
For u,v contained in R^n and u,v not equal to zero...

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University of Maryland
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