[理工] 線代
1. 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) Show that W3 is a subspace of R^n.
(b) 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) 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) 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) For the subspace W1 and W2 defined in (d), find a basis for
交集(W1, W2).
2Let A be a 3×3 matrix such thea A^-1 = -A. Show that whether A is
invertible?
3Let 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.
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 ] }
請問這幾題要如何解決 謝謝..
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