課程名稱︰線性代數
課程性質︰必修
課程教師︰蘇柏青
開課學院：管理學院
開課系所︰工管系科管組
考試日期（年月日）︰104/10/30
考試時限（分鐘）：50 min
試題 :
Linear Algebra Quiz #1
1. Properties of Matrices(80%)
- -
| 1 1 3 0 1 |
| 1 1 3 0 1 |
Consider the matrix A = | 0 0 0 1 -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 1 2 3] 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%)
(a) (10%) Let S={u1,u2,…,uk} be a finite subset of R^n. Prove that if u and
v are both in Span S, then u+cv is also in Span S for any scalar c.
(b) (10%) Let u and v be distinct vectors in R^n. Prove that the set {u,v} is
linearly independent if and only if the set {u+v,u} is linearly independent.