[試題] 96下 電機系統一教學 線性代數 期中考
課程名稱︰線性代數
課程性質︰大一必修
課程教師︰
開課學院：
開課系所︰電機系
考試日期（年月日）︰96/04/16
考試時限（分鐘）：100分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. (21%) Label the following statememts as being true or false. (Explain your
answer. Answers with no explanation get 0%):
(a) Let A be an m × n matrix with reduce row echelon form R. If rank A = N
then there is a unique matrix P such that PA = R.
(b) If (A^T)A = O, then A = O.
(c) If A is an n × n matrix and the system Ax = b is consistent for every
b, then A is invertible.
(d) Let S be a linearly independent subset of R^n and V be a k-dimensional
subspace of R^n. If S has k vectors, then S is a basis of V.
(e) If a linear transformation is one-to-one, then it is invertible.
(f) Let V and W be two subspace of R^2. V∪U is a subspace of R^2.
(g) Let R be the reduced row echelon form of A. Then R and A have the same
column space.
2. (9%) Find a basis for the column space and the null space of the following
matrix.
[ -1 2 1 -1 -2 ]
[ 2 -4 1 5 7 ]
[ 2 -4 -3 1 3 ]
3. Given the matrix a below, find the determinants of (a) (8%) A, (b) (2%)
A^(-1), (c) (3%) 2A, and (d) (2%) R where R is the reduced row echelon form
of A.
[ 0 -1 2 -1 ]
A = [ 1 0 -1 1 ]
[ 1 -1 1 -1 ]
[ -1 1 1 0 ]
4. (10%) Let V and W be two subspaces of R^n. Show whether
V + W = {sεR^n : s = v+ w, vεV and wεW}
is a subspace of R^n.
5. Let T: R^3 → R^3 be the linear transformation defined by
( [ x1 ] ) [ x1 ﹣x2 + x3 ]
T ( [ x2 ] ) = [ -x1 + x2 ﹣3x3 ]
( [ x3 ] ) [ 2x1 + x3 ]
(a) (2%) Find the standard matrix of T. (b) (4%) Is T one-to-one? Is T
onto? (c) (9%) Is T invertible? If it is, find T^-1.
6. (a) (5%) Let A be a 3×3 matrix such thea A^-1 = -A. Show that whether A is
invertible?
(b) (5%) Let C be a 3×3 matrix with det C = 2. What is the determinant of
B = [ C -C ]?
[ C C ]
7. Let V1 be a 3-dimensional subspace of R^4 and V2 be a 2-dimensional subspace
of R^4. Let W be the intersrction of V1 and V2, i.e., W = V1∩V2.
(a) (6%) Prove that W is also asubspaceof R^4.
(b) (8%) Prove that W contains some nonzero vectors.
(c) (6%) Let V1 and V2 be respectively the span of S1 and S2 given below.
Find a basis for W.
{ [ 1 ] [ -1 ] [ 2 ] } { [ 1 ] [ 2 ] }
S1 = { [ -1 ],[ 1 ] [ 2 ] }, S2 = { [ 3 ],[ 6 ] }
{ [ 2 ] [ -3 ] [ 2 ] } { [ 0 ] [ 3 ] }
{ [ -3 ] [ 2 ] [ 0 ] } { [ 3 ] [ 9 ] }
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04/16 00:23, , 1F
04/16 00:23, 1F