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課程名稱︰線性代數
課程性質︰
課程教師︰馮世邁
開課系所︰電機系
考試時間︰94/04/19
試題 :
1. Let A be the matrix given below.
[ 1 2 2 -1 2 ]
A = [a1 a2 a3 a4 a5] = [-1 2 -2 1 6 ]
[ 2 2 2 -3 3 ]
[-3 0 -6 2 9 ]
(a) (5%)Find the reduce row echelon form of A.
(b) (5%)Write all the non pivot columns of A as a linear combination of
pivot columns of A.
(c) (5%)Find a basis for Col A, Row A, and Null A respectively.
(d) (10%)Determine if the set S given below is a basis for Col A.
{[ 2] [ 2] [ 4]}
S = {[ 2], [ 2], [ 0]}
{[ 1] [ 3] [ 7]}
{[-1] [ 1] [-5]}
2. Let T be a linear operator on R^3 and T(v1) = w1, T(v2) = w2, T(v3) = w3,
where the vactors are defined by
[ 0] [ 2] [ 1] [-1] [ 1] [ 1]
v1 = [ 1], v2 = [ 0], v3 = [ 2], w1 = [ 1], w2 = [-1], w3 = [ 1]
[ 2] [ 0] [ 0] [ 1] [ 1] [-1]
(a) (5%)Let C = [v1 v2 v3]. Find det C.
(b) (5%)Is C invertible? If it is, find its inverse.
(c) (5%)Is T uniquely determined? If it is, find its standard matrix.
(d) (5%)Is the set B = {v1, v2, v3} a basis for R^3? If it is, find [T]_B.
3. Let [R c] be the reduced row echelon form of [A b]. Show that Rx = c is
equivalent to Ax = b.
4. Let U: R^2 -> R^2 be the orthogonal projection onto the line y = mx.
Find U ([ x]).
([ y])
5. Let W1 and W2 be subspace of R^n and let W3 = {z belongs to R^n: z = u + v
for some u belongs to W1, v belongs to W2}.
(a) (5%)Show that W3 is a subspace of R^n.
(b) (5%)Show that any subspace of R^n containing W1 and W2 will also
contain W3, i.e., W3 is the smallest subspace of R^n contains
W1 and W2.
(c) (5%)Let S1 and S2 be spaning sets of W1 and W2 respectively. Show that
there is a basis of W3 that is contained in 聯集(S1, S2).
(d) (3%)Suppose W1 = Span{a1, a3, a5} and W2 = Span{a2, a4}, where the
vactors a_i are defined in Eqn. (1) of Problem 1. Find a basis for
W3.
(e) (7%)For the subspace W1 and W2 defined in (d), find a basis for
交集(W1, W2).
6. (a) (5%)Fond a 2*2 matrix A such that Col A = Null A.
(b) (5%)Can we find a 3*3 matrix B such that Col B = Null B? Justift
your answer
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