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“There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of pro- fessors and textbook writers have obscured its simplicity by preposterous calculations with matrices.” –J. Dieudonne
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Questions begin on the next page. Please answer each question in the spaces provided; if you need extra room, you can attach extra blank pages at the end of the worksheet.
The worksheet follows the standard class collaboration policy. You may ask other students, the instructor, or the TA for help, but all work written on this worksheet must be entirely your own. Do not simply copy somebody else’s answers.
You may look at linear algebra textbooks, online tutorials, math.stackexchange, or any other resources to help you complete the worksheet. Do not worry if you cannot do the problems from memory alone! You are allowed, and expected, to look up any formulas or definitions as needed.
The problems are not intended to be difficult or time-consuming to solve, but rather, to refresh your memory on basic linear algebra concepts, and to give you a self-check on the prerequisites you will need to succeed in this class. If you get stuck on any problem, the TA or instructor would be happy to help you during office hours.
Useful Formulas
- Dot product: (ux, uy , uz , uw) · (vx, vy , vz , vw) = uxvx + uy vy + uz vz + uwvw.
- Cross product: (ux, uy , uz ) × (vx, vy , vz ) = (uy vz − uz vy , uz vx − uxvz , uxvy − uy vx).
- Norm: ‖u‖ =
u · u =
uT^ u.
- Unit vector: ˆu = (^) ‖uu‖.
- Angles between vectors: u · v = ‖u‖‖v‖ cos θ, ‖~u × ~v‖ = ‖u‖‖v‖| sin θ|.
- Recall that a function f (v) of a vector v is linear if f (αv) = αf (v) and f (v + w) = f (v) + f (w). For each of the following functions, (a) state whether or not the function is linear, and (b) if the function is not linear, give a counterexample where the function violates one of the above properties. (a) (1 point) Rescaling of the input vector, f (v) = 5v.
(b) (1 point) The constant function f (v) = 2.
(c) (1 point) Translation f (v) = v + w for some fixed vector w 6 = 0.
(d) (1 point) The function returning the first coordinate of a vector, f
v 0 v 1 v 2
(^) = v 0.
(e) (2 points) Multiplication by a fixed matrix Mn×m, f (v) = M v, where v ∈ Rm.
(f) (2 points) f (v) = w · v, for a fixed vector w.
(g) (2 points) The norm function f (v) = ‖v‖.
- Let w ∈ R^2 be a fixed vector. Let
f (v) =
I −
wwT ‖w‖^2
v,
for vectors v ∈ R^2 in the plane, where I =
[
]
is the identity matrix.
(a) (2 points) Is this function linear? If not, provide a counterexample.
(b) (2 points) Compute f (w).
(c) (2 points) Compute f (v), where v is any vector perpendicular to w (i.e. v · w = 0).
(d) (4 points) Describe using words, and no equations, what the function f does to vectors. To build intuition, it may help to try concrete numerical examples, or to draw a picture.
- A key skill in linear algebra is turning systems of linear equations into matrix equations of the form Ax = b, where A and b are known, and x is unknown. (a) (5 points) Let v 1 , v 2 , v 3 and w be four different vectors in R^3. You are told that
w = αv 1 + βv 2 + γv 3
for some unknown real numbers α, β, and γ. Write a matrix equation (of the form Ax = b) that would allow you to solve for these unknown scalars.
(b) (5 points) Let v 1 , v 2 , v 3 be three different vectors in R^3 , and k 1 , k 2 , k 3 three real numbers. You are told that
k 1 = w · v 1 k 2 = w · v 2 k 3 = w · v 3
for some unknown vector w ∈ R^3. Write down a matrix equation that would allow you to solve for this vector w.
