Ma 125 Written Homework 3 Due: March 7, 2025
Write solutions to the following questions, then take clear pictures or scans to upload on
Gradescope. Sufficient steps shown, organization of the solution, correct mathematical
notation and clearly stated reasoning where appropriate are all necessary.
1. Let
1 1 1 −1
v1 = 1 , v2 = 2 , v3 = −3 , v4 = 1
−2 −2 −2 2
1
(a) Is the vector u = 2 contained in Span{v1 , v2 , v3 , v4 }? If so, write u as a linear
−1
combination u = a1v1 + a2v2 + a3v3 + a4v4 using as few non-zero scalars for
a1 , a2 , a3 , a4 as possible.
2
(b) Is the vector w = −3 contained in Span{v1 , v2 , v3 , v4 }? If so, write u as a linear
−4
combination w = b1v1 + b2v2 + b3v3 + b4v4 using as few non-zero scalars for a1 , a2 , a3 , a4
as possible.
2. Find the largest possible number of linearly independent vectors among
1 1 1 0 0 0
−1 0 0 1 1 0
v1 =
0 , v2 = −1 , v3 = 0 , v4 = −1 , v5 = 0
, v6 =
1
0 0 −1 0 −1 −1
In other words, what’s the size of the largest linearly independent set formed from the
vectors above? Justify your answer.
3. Let {u1 , u2 , u3 } be a set of linearly independent vectors, and define the following vectors.
v1 = u2 + u3 , v2 = u1 + u3 , v3 = u1 + u2
1 = u2 − u3 , w
w 2 = u1 − u3 , w
3 = u1 − u2
(a) Determine whether {v1 , v2 , v3 } are linearly independent or dependent.
(b) Determine whether {w
1, w 3 } are linearly independent or dependent.
2, w
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Write solutions to the following questions, then take clear pictures or scans to upload on
Gradescope. Sufficient steps shown, organization of the solution, correct mathematical
notation and clearly stated reasoning where appropriate are all necessary.
1. Let
1 1 1 −1
v1 = 1 , v2 = 2 , v3 = −3 , v4 = 1
−2 −2 −2 2
1
(a) Is the vector u = 2 contained in Span{v1 , v2 , v3 , v4 }? If so, write u as a linear
−1
combination u = a1v1 + a2v2 + a3v3 + a4v4 using as few non-zero scalars for
a1 , a2 , a3 , a4 as possible.
2
(b) Is the vector w = −3 contained in Span{v1 , v2 , v3 , v4 }? If so, write u as a linear
−4
combination w = b1v1 + b2v2 + b3v3 + b4v4 using as few non-zero scalars for a1 , a2 , a3 , a4
as possible.
2. Find the largest possible number of linearly independent vectors among
1 1 1 0 0 0
−1 0 0 1 1 0
v1 =
0 , v2 = −1 , v3 = 0 , v4 = −1 , v5 = 0
, v6 =
1
0 0 −1 0 −1 −1
In other words, what’s the size of the largest linearly independent set formed from the
vectors above? Justify your answer.
3. Let {u1 , u2 , u3 } be a set of linearly independent vectors, and define the following vectors.
v1 = u2 + u3 , v2 = u1 + u3 , v3 = u1 + u2
1 = u2 − u3 , w
w 2 = u1 − u3 , w
3 = u1 − u2
(a) Determine whether {v1 , v2 , v3 } are linearly independent or dependent.
(b) Determine whether {w
1, w 3 } are linearly independent or dependent.
2, w
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