Math 51, Spring 2011 Henry Adams, April 18

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Math 51, Spring 2011
Henry Adams, April 18
These are the problems we solved during my review session for midterm 1.
(1) Suppose v ∈ Rn is a unit vector (that is, kvk = 1). Prove that for any vector w ∈ Rn , the vector
w − (w · v)v is orthogonal to v.
(2) Let T : Rn → Rn be a linear transformation and let V = {x ∈ Rn | T (x) = 5x}. Prove that V is a
linear subspace of Rn .
(3) Suppose v1 , v2 , and v3 are nonzero vectors that are orthogonal to each other. Prove that the set
{v1 , v2 , v3} is linearly independent.

3 0 −3 12 0
1 2 3 10 0

(4) Let A = 
2 1 0 11 1.
0 1 2
3 1
(a) Find rref(A).
(b) Find nullity(A) and rank(A).
(c) Find N (A) and a basis for N (A).
(d) Find C(A) and a basis for C(A).
 
1
2
 

(e) Find all solutions x to Ax = Ac, where c = 
3.
4
5
(f) Is there a vector b ∈ R4 such that Ax = b has no solutions x?
(g) Is there a vector b ∈ R4 such that Ax = b has exactly one solution x?
(5) Consider the following linear subspace of R4 .
 
( x1 )
x2  x1
−2x3 +3x4 = 0


V =  
x3 x2 −x3
=0
x4
(a) Find a matrix A such that N (A) = V .
(b) Find
B such
 C(B) = V .

 that
 a matrix
−2
4
(6) Let v = −1 and w =  2 .
1
1
(a) Find the angle between v and w.
(b) Find the area of the triangle with vertices v, w, and ~0.
(7) Let {u, v, w} be a basis for a subspace V of Rn . Is {u − v, v − w, u − w} a basis for V ?
(8) Let T : R2 → R2 be the linear transformation that satisfies
 
 
−3
0
1/2
1/2 T
=  1  and T
= 1 .
1/2
−1/2
0
2
Find matrix A such that T (x) = Ax for all x ∈ R2 .
(9) Consider the system of equations
x − 3y = b1
3x + ay = b2
where b1 and b2 are fixed constants. For which values of a does the system of equations have exactly
one solution?
1
 
1
(10) Find a parametric equation for the plane in R3 that passes through the points 2,
3
2
 
 
4
7
5, and  8 .
6
10
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