Math 221 Midterm 1 practice(100 points)
Thur. Oct. 2, 2025 8:30am-9:45am
Name:
The exam is closed-book. Please note that no use of a computer (including Python or
any coding language), internet, or cell phone is allowed, nor discussion with anyone in any
form.
The exam time is 75 minutes for you to work on the questions and to check the answers. The total time for you from looking at the problems to ending writing
should be no longer than 75 minutes.
In written problems, if no derivation is asked for, just give the answer.
give the derivation in a clear and mathematically rigorous way.
Otherwise,
1. (20 points) True or False.
(1)
(2)
(3)
(4)
(5)
3
1
, compute x · y and the angle θ between them.
and y =
2. (5 points) For x =
4
2
Math 221 Midterm 1 practice
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3. (10 points) The equation 2x1 − 3x2 − 6x3 = −4 defines a plane in R3 .
(a) Give its normal vector a.
(b) Find the distance from the origin to the plane.

  
0
1




1
, 2 .
4. (15 points) Find all vectors x ∈ R4 that are orthogonal to the hyperplane spanned by 
−1  0 
2
−2
Math 221 Midterm 1 practice
5. (10 points) Use vector methods to prove that (some geometric properties)
6. (10 points) Let v, w ∈ Rn . Prove that span{v, w} = span{v + w, v} .
7. (10 points) Find the maximum of x + 2y + 3z given 3x2 + 4y 2 + 5z 2 = 1
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Math 221 Midterm 1 practice
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8. (10 points) Let T : R2 → R2 be the linear transformation defined by (some descriptions of the
transformation)
Give the standard matrices representing T .
Math 221 Midterm 1 practice
9. (10 points) Some proof questions
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