Algebra homework help

Algebra homework help. Written Assignment 4
Due: Wednesday, March 18 (in class, before the lecture begins).
Instructions: Attempt all questions. You should provide appropriate justification for
your answers and refrain from using formulae/results that lie outside the course content;
unsubstantiated answers will not receive full credit.
1. [10] Let L: R
2 → R
2 be the linear operator that reflects vectors about the line with
equation ax + by = 0, where a and b are unspecified real numbers (with ab 6= 0).
(a) [5] Find a formula for the standard matrix [L] (the entries of your matrix
should be expressed in terms of a and b).
(b) [5] Give a geometric explanation of why [L] is invertible, and then find [L]
−1
.
2. [10] Let
A =






1 1 2 −5 −1
0 3 −6 7 3
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0






(a) [2] Find a basis of the row space of A.
(b) [5] Find a basis of the null space of A.
(c) [3] Find rank(AT
), and then find nullity(AT
).
3. [10] Consider the vectors
~v1 =




1
2
−1
1




, ~v2 =




1
3
−1
1




, ~v3 =




8
19
−8
8




, ~v4 =




−6
−15
6
−6




, ~v5 =




1
3
1




, ~v6 =




1
5
1




(a) [7] Find, with justification, a subset of {~v1, ~v2, ~v3, ~v4, ~v5, ~v6} that forms a basis
of the subspace S of R
4
spanned by these six vectors.
(b) [3] Express each of the vectors ~v1, ~v2, ~v3, ~v4, ~v5, ~v6 as a linear combination of the
elements of the basis you found in (a). Justify your answers.
4. [8] Let A and B be n × n matrices.
(a) [4] Show that Null(B) ⊆ Null(AB).
(b) [4] Show that if A is invertible, then the reverse inclusion Null(AB) ⊆ Null(B)
also holds (and so Null(AB) = Null(B)).
5. [14] Consider the 3 × 3 matrix
A =


1 0 3
2 3 4
1 0 2


(a) [6] Show that A is invertible and compute its inverse A−1
.
(b) [4] Express A as a product of elementary matrices.
(c) [4] Use A−1
to solve the system of equations
x + 3z = a
2x + 3y + 4z = b
x + 2z = c
where a, b, and c are unspecified real numbers (no points will be awarded if
A−1
is not used).
6. [8] Let ~xT = (x1, x2, …, xn) be a row vector in R
n
, and A be an n×m matrix. Show
that ~xTA is a linear combination of the rows of the matrix A.

Algebra homework help