課程名稱︰線性代數
課程性質︰電機系必修
課程教師︰劉志文
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰108.11.08
考試時限（分鐘）：100
試題 :
註：以下的屬於符號皆以belong to 代替。
1.(16%) Let V and W be subspace of R^n
(a)(8%) If V∪W is a subspace of R^n, then prove it. Otherwise, give a counte-
rexamople.
(b)(8%) Let V + W = { v + w : v belong to V and w belong to W }. If V + W is a
subspace of R^n, then prove it. Otherwise, give a counterexample.
┌1 2 3┐ ┌2┐
2.(10%) Let A = │3 4 8│and b =│1│. Find A^{-1}b.
└2 5 6┘ └3┘
┌ 1 2 -1 2 1┐
3.(14%) Let A = │-1 -2 1 2 1│.
│ 2 4 -3 2 0│
└-3 -6 2 0 3┘
(a)(8%) Find the reduced row echelon form of A.
(b)(6%) Let { b1, b2, ... , bi } be a basis of Null A. Let B = [b1, b2,..., bl
]. Find the dimension of Null B^T.
4.(10%) Let βbe a basis of R^n. Prove that the function ψ: R^n → R^n defin-
ed by ψ(v) = [v]_β(以β為底) is a one-to-one and onto linear transformation,
where [v]_βis the coordinate of v relative to β.
5.(13%)
(a)(7%) Let Q be an nxn invertible matrix. Let { v1, v2, ... , vk } be a line-
arly independent set of vectors in R^n. Prove that {Qv1, Qv2, ... , Qvk } is
linearly independent.
(b)(6%) Suppose that { u1, u2, ... , un } and { Pu1, Pu2, ... , Pun } are both
linearly independent subsets of R^n, where P is an nxn matrix. Prove or dispr-
ove that P is an invertible matrix.
┌1 2c 1-c┐
6.(6%) Let A = │0 0 -1-c│. Calculate det A in terms of c. Determine that
└1 -1+c 3┘
conditions on c so that A is invertible.
7.(10%) Consider the following linear transformations:
┌x1┐ ┌ x1 + 2x2┐
T1 : R^2 → R^3 defined by T1(│ │) = │-x1 + 3x2│
└x2┘ └-x1 - x2┘
┌x1┐ ┌ x1 + x2 + 2x3┐
T2 : R^3 → R^2 defined by T2(│x2│) = │ │
└x3┘ └ x1 - x2 ┘
(a)(8%) Let matrices A, B, C and D be the standard matrices of T1, T2, T1。T2
and T2。T1, respectively. Find A, B, C and D.
(b)(4%) Is T1。T2 onto? Is T1。T2 one-to-one?
(c)(4%) Is T2。T1 onto? Is T2。T1 one-to-one?
8.(15%) Let A be an mxn matrix, and let E1 and E2 be an mxm elementary matrix
and an nxn elementary matrix, respectively. Determine the validity of each of
the following statements. You don't have to give a proof if you determine a
statement is true; however, if yoy think a statement is false, find a counter-
example to show it. (i.e. a specific example of matrices A and E1 (or E2).)
(a)(3%) Col A = Col E1A.
(b)(3%) Row A = Row E1A.
(c)(3%) Col A = Col AE2.
(d)(3%) Null A = Null AE2.
(e)(3%) Null A = Null E1A.