**Unformatted text preview: ** and .
x
y
b) Determine the value of · .
xy 1
1
5. Let P be the plane in R3 with vector equation = s 4 + t 2.
x
1
0
a) Find a scalar equation for P of the form ax1 + bx2 + cx3 = d. 1
b) Find the projection of = 0 onto P .
x
0
1 3
1
5
1 , −1 , 1 represents a line or a plane
6. Determine if the span of 0
2
−1
in R3 and give a simpliﬁed vector equation which describes it. 7. Determine, with proof, which of the following are subspaces of R3 and which are not. If
the set is a subspace, then ﬁnd a basis for it.
(a) The solution set of the system of equations
3 x1 + 4 x2 − x3 = 5
2 x1 − 3 x2 + 2 x3 = 6 x1 (b) The set S = x2 ∈ R3 | x1 = x2 − x3 . x3 8. Prove that if 1 , 2 ∈ Rn and s, t are non-zero real numbers, then
vv
span{1 , 2 } = span{1 , s1 + t2 }
vv
vv
v
9. Let S = {1 , . . . , k } be a set in Rn .
v
v
a) Prove that if S is linearly independent, then k ≤ n.
b) Prove that if span S = Rn , then k ≥ n.
10. Let B =...

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