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課程名稱︰線性代數
課程性質︰電機系必修
課程教師︰蘇柏青
開課學院：電資學院
開課系所︰電機工程學系
考試日期（年月日）︰2015/4/22
考試時限（分鐘）：100 分鐘
試題 :
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LINEAR ALGEBRA: Midterm Examination 2015
1. (10%) Let A be an n x n matrix. Let R be the reduced row echelon form of A.
Prove or disprove that det(A) = det(R).
2. (10%) Let T: R^n → R^m be a linear transformation defined by T(x) = Ax for
all x ∈ R^n, where A is an m x n matrix. Let U: R^m → R^n be a linear
transformation defined by U(x) = A^T x for all x ∈ R^n, where A^T is the
transpose of A. Prove or disprove that the dimension of the range of T is
the same as the dimension of the range of U.
3. (a)(6%) Let Q be an n x n invertible matrix. Let {u1,u2,…,uk} be a linearly
independent set of vectors in R^n. Prove that {Qu1,Qu2,…,Quk} is linearly
independent.
(b)(4%) Suppose that {Pu1,Pu2,…,Puk} is linearly independent, where P is an
n x n matrix. Is it necessary that k <= n?
4. (15%) Let A = [a1 a2 a3 a4 a5] be a 4 x 5 matrix and b ∈ R^4. The general
solution to Ax = b is given by
x1 -5 -2 1
x2 0 1 0
[ x3 ] = [ -3 ] + x2 [ 0 ] + x5 [ 0 ]
x4 2 0 -1
x5 0 0 1
(a)(3+2%) Find that rank and nullity of A.
(b)(3%) Find a basis for Null A.
(c)(7%) Let A' = [a2 a1 a3 a4 -a5] and b' = b + a3. Find the general
solution to A'x = b' in vector form.
5. (20%) Consider the following linear transformation:
x1 x1 + x2
T1: R^2 → R^3 defined by T1( [ x2 ] ) = [ x1 - 3x2 ]
4x1
x1 x1 - x2 + 4x3
T2: R^3 → R^2 defined by T2( [ x2 ] ) = [ x1 + 3x2 ]
x3
(a)(8%) Find the standard matrices of T1,T2,T1T2 and T2T1.
(b)(4%) Is T1T2 onto? Is T1T2 one-to-one?
(c)(4%) Let U1: R^n → R^m and U2: R^m →R^p be linear. Determine if the
following statements are true or false(explain your answer):
(i) If m > n, then U2U1 cannot be onto.
(ii) If m < p, then U2U1 cannot be onto.
(d)(4%) Let U1 and U2 be linear transformation defined in (c). Prove that if
U1 and U2 are one-to-one, then U2U1 is one-to-one.
1 1 1
6. (20%) Consider the 4 x 3 matrix A = [ 1 a b ] where a,b ∈ R.
1 a^2 b^2
1 a^3 b^3
(a)(8%) Show that the column vector of A form a linearly independent set if
and only if a ≠ 1, b ≠ 1, and a ≠ b.
(b)(6%) Assume that a = b ≠ 1. Find the rank A and nullity A.
(c)(6%) Assume a ≠ 1, b ≠ 1, and a ≠ b. Let b = [0 1 a+b a^2+ab+b^2]^T
∈ R^4. Show that b ∈ Col A by explicitly specifying a vector x ∈ R^3
such that Ax = b.(Hint: Recall that a^3 - b^3 = (a-b)(a^2+ab+b^2) and
a^2 - b^2 = (a-b)(a+b).)
7. (15%) Let A be an m x n matrix and R be its reduced row echelon form. We
know that there is an m x m invertible P such that PA = R.
(a)(8%) Use the formula PA = R to prove that the rows of R are linearly
independent if and only if the rows of A are linearly independent.
(b)(7%) If rank A = m, show that P is uniquely determined by proving that
P = [ a_p1 a_p2 … a_pm ]^-1, where a_p1,a_p2,…,a_pm are the pivot
columns of A.
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