Math 340A (Summer 2016)
Quiz 1 Solutions
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Student ID:
1. Complete the following definitions. See the example.
Example: A vector space V is finite-dimensional if it has a finite basis.
(a) The trace of a matrix A Fnn is the sum of diagonal entries

Math 340A (Summer 2016)
Homework 2: Solutions
1. Read Examples 18 and 19 in FIS, page 50. Solve: FIS, page 56, problem 15.
Solution. A basis for W is given by cfw_E ij : 1 i, j n, i 6= j cfw_Ai : 1 i n 1, where
Ai Fnn is a diagonal matrix such that Aiii =

Math 340A (Summer 2016)
Homework 3: Solutions
1. FIS, page 75, problem 6.
Solution. To prove the trace is linear, we let A, B Fnn and , F. Then:
tr(A + B) =
n
n
n
n
X
X
X
X
(A + B)ii =
Aii + Bii =
Aii +
Bii = tr(A) + tr(B).
i=1
i=1
i=1
i=1
For any F, co

16
HON LEUNG LEE
Among all subspaces that contain S, span(S) is the smallest one.
Theorem 1.14. span(S) is the smallest subspace of V containing S.
Proof. The subspace cfw_0V is the smallest one containing . Next, we suppose S is nonempty.
S span(S) V :

14
HON LEUNG LEE
Next we look into subspaces. The definition is very similar to that in Math 308.
Definition 1.7. Let V be a vector space over F with the operations +, . A subset W of V is a
subspace of V if: 0V W , and, for any x, y W , F, one has x + y

CHAPTER 1
Vector Spaces
1.1. Vector spaces and subspaces
[FIS, 1.2,1.3]
Symbols: means for all, means there exists (at least one), ! means unique, := means
is defined to be.
In Math 308 we learnt vectors in Rn . We can add two vectors in Rn , and multiply

MATH 340 LINEAR ALGEBRA
17
We are afraid that given a vector space V and two bases 1 , 2 , it may happen that |1 | = 3 but
|2 | = 8. Then the dimension of V does not make any sense. We are going to prove this cannot
happen. This is the first long proof we

Chapter 4, Lesson 1: Protons, Neutrons, and Electrons
Key Concepts
Atoms are made of extremely tiny particles called protons, neutrons, and electrons.
Protons and neutrons are in the center of the atom, making up the nucleus.
Electrons surround the