Unformatted text preview: Math 601 HOMEWORK 4 1. For each of the following matrices 12
, B = 0 0 , C =
a) ﬁnd the null space.
b) Find the column space.
c) Is the matrix invertible? If so, ﬁnd its inverse, and check your answer.
d) Does the matrix have a left inverse? If so, ﬁnd it, and check your answer.
e) Does the matrix have a right inverse? If so, ﬁnd it, and check your answer.
2. Construct a matrix which transforms the standard basis vectors e1 , e2 , e3
of R3 into three given vectors v1 , v2 , v3 ∈ R3 . When is this matrix invertible?
3. Describe the linear transformation of the plane R2 which are represented in the standard basis by the matrices
A1 = 0 −1
10 −1 0
01 , A2 = , A3 = 01
00 4. Let A be an n × n (square) matrix.
a) Show that the null space of A2 contains the nulls pace of A.
b) Show that the column space of A2 is contained in the column space of A. 1 ...
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