课程名称︰线性代数 Linear Algebra
课程性质︰电机系必修
课程教师︰苏柏青
开课学院：电资学院
开课系所︰电机系
考试日期（年月日）︰2015/3/26
考试时限（分钟）：50 min
试题 :
Linear Algebra Quiz #1
1. Properties of Matrices(80%)
- -
| 0 0 0 1 -1 |
| 0 2 2 0 0 |
Consider the matrix A = | 1 0 2 2 -1 |
| 0 1 1 0 0 |
- -
(a) (4%) How many columns does A have? How many rows does A have?
(b) (10%)Find the reduced row echelon form R for the matrix A using Gaussian
Elimination. (Keep track of each elementary row operation you performed to
obtain R for future use)
(c) (8%) For each elementary roe operation you did in (b), write the correspon
ding elementary matrix.
(d) (5%) Find an invertible matrix P such that R = PA (Hint:Use results in (b))
(e) (10%)Write P as a product of several elementary matrices.
(f) (6%) What is the rank of A? What is the nullity of A?
(g) (3%) Which columns of R are pivot columns?
(h) (9%) Let ai denote the ith column of A. Is {a1,a2} linearly independently?
Is{a1,a2,a3} linearly independent? Is {a1,a2,a4} linearly independent?Why?
T
(i) (7%) Is the vector b =[ 1 2 3 4] in the span of all columns of A?
Δ
(j) (6%) Let T be a linear transformation induced by matrix A, i.e,T(x) = Ax.
What is the domain of T? What is the codomain of T?
(k) (6%) What is the range of T? Is T an onto transformation?
(l) (6%) Find the null space of T (i.e., the set of all v in the domain suth
that T(v) = 0). Is T a one-to-one transformation?
2. Proof of theorems(20%)
Consider an m x n matrix A = [a1 a2 ... an] and let its reduced row echelon
form be R = [r1 r2 ... rn]. Form the class we learned that there exist an
invertible matrix P of size m x m such that PA = R.
(a) (8%) Suppose a subset of column vectors of A, {ap1,ap2, ... apk},is
linearly independent. Here k is an integer,k <= n,and 1 <= p1 < p2 <...< pk
<=n.Show that {rp1,rp2, ... rpk} is also linearly independent.
m
(b) (12%)Show that Ax = b is consistent for all b ∈ R if and only if rank A =
m ("if": 6%; "only if": 6%).