Q2a) Consider a transformation T : R 2×2 → R 2×2 such that T(M) = MT...

Q2a) Consider a transformation T : R
2×2 → R
2×2
such that T(M) = MT
.
This is infact a linear transformation. Based on this, justify if the following
statements are true or not. (2)
a) T ◦ T is the identity transformation.
b) The kernel of T is the zero matrix.
c) Range T = R
2×2
d) T(M) =-M is impossible.
b) Assume that you are given a matrix A = [aij ] ∈ R
n×n with (1 ≤ i, j ≤ n)
and having the following interesting property:
ai1 + ai2 + ..... + ain = 0 for each i = 1, 2, ...., n
Based on this information, prove that rank(A) < n. (2)
c) Let A ∈ R
m×n be a matrix of rank r. Suppose there are right hand sides
b for which Ax = b has no solution, which of following expression(s) is/are
correct: r < m, r = m, r > m.
Now, consider the linear system AT y = 0. Do you think this linear system can
have non-zero solutions, that is y 6= 0 such that AT y = 0. Give justification for
all your answers.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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