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.