Linear Algebra
MATH 211 Spring 2020
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: R2 → R2 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 0 1
,~v6 =
1 5 0 1
(a) [7] Find, with justification, a subset of {~v1,~v2,~v3,~v4,~v5,~v6} that forms a basis of the subspace S of R4 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 32 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 Rn, and A be an n×m matrix. Show that ~xT A is a linear combination of the rows of the matrix A.