MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 1; Solutions
[1] Since we are dening the operations of addition and multiplication on A[ 2] to be those inherited
from R, we need in particular to check that, with these operations, the set A[ 2] is c
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 4; Solutions
[1] Let V be a vector space over a eld F . Suppose that the subset cfw_u, v, w is a linearly independent
subset of V .
[a] Suppose F = C is the eld of complex numbers. We want to prove th
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 3; Solutions
[1] [a] Let F be a eld, and let A and B be two 2 2 matrices over F . If AB = I , where I is the 2 2
identity matrix, prove that BA = I as well.
Proof: One way is to use the theorem we pro
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 6; Solutions
[1] Let T : R3 R3 be the mapping dened by
T (x1 , x2 , x3 ) = (x1 + 2x2 + x3 , 2x1 x2 + x3 , x1 + 2x2 + x3 ).
[a] Let x = (x1 , x2 , x3 ) and y = (y1 , y2 , y3 ) be in R3 . We can compute
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 2; Solutions
[1] Let the eld be C. Let
1
B = 1
9
3
1
2
3
0
1
2
6
We will solve both problems [a] and [b] at once by row-reducing the
possible sequence of elementary row operations is the following:
.
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 5; Solutions
[1] Given any vector v = (v1 , v2 , v3 ) R3 , dene a matrix Av by
0
v3 v2
0
v1 .
Av = v3
v2 v1
0
Given v, w R3 , dene B (v, w) = v1 w1 + v2 w2 + v3 w3 . For any v R3 , dene two subsets Sv
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 7; Solutions
[1] [a] Suppose V and W are both nite-dimensional. If dim V = dim W = n, then we proved in class
that both V and W are isomorphic to F n , hence they are isomorphic to each other. Now sup
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 8; Solutions
[1] Consider a sequence of length k .
[a] If k = 1, the sequence is of the form 0 V1 0, where both maps and are necessarily zero
maps. Exactness at V1 is equivalent to the ker = im . But
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 12; Solutions
[1] Let V be an n-dimensional vector space, and let T : V V be a linear operator. Suppose W is a
T -invariant subspace of V . We proved in class that the minimal polynomial q of T |W div
MATH 146: Linear Algebra 1 (Advanced Level)
Mid-term Test; Solutions
[1] State whether each of the following statements is TRUE or FALSE. Give a short (one or two sentences)
justication for each of your answers. The correct answer with the wrong justicati
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 9; Solutions
[1] We consider the following expressions which dene functions D on the set of 3 3 matrices over Q.
Recall that a function on 3 3 matrices is called 3-linear if, when two of the rows are
MATH 146: Linear Algebra 1 (Advanced Level)
Assignment 11; Solutions
[1] Suppose that A is a 2 2 matrix with real entries that is symmetric (AT = A). Therefore A is of the
form
ab
A=
,
a, b, c R.
bc
Its characteristic polynomial is thus
p = det(xI A) = de