MATH 232 Page 1 of 14
1. Consider the plane in R3that goes through the point (2,1,3) and is perpendicular to the
vector (1,1,1).
[3] (a) Give a general equation of this plane.
[3] (b) Give a vector equation of this plane.
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2nd midterm for linear algebra
Typology: Study notes
2024/2025
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1. Consider the plane in R^3 that goes through the point (2, 1 , 3) and is perpendicular to the
vector (1, 1 , 1).
[3] (a) Give a general equation of this plane.
[3] (b) Give a vector equation of this plane.
2. Consider the matrix A =
[4] (a) Find elementary matrices E 1 and E 2 such that E 2 E 1 A = I.
[2] (b) Write A−^1 as a product of two elementary matrices.
[3] (c) Write A as a product of two elementary matrices.
[4] (c) If we let the basis B = {v 1 , v 2 , v 3 }, what is [ProjW (3, 3 , 3 , 3)]B , that is, the coordinates of the projection of (3, 3 , 3 , 3) onto W with respect to the basis B?
[3] (d) What is a basis for W ⊥?
[5] 4. Let A and B be two square matrices of the same size. Suppose B is singular. Explain why
AB is also singular.
- [5] (b) Rotation about the origin counter-clockwise by π/
6. Suppose the transition matrix for a Markov process is
State A State B State A State B
[
1 − p 1 p 0
]
where 0 < p < 1. So, for example, if the system is in state A at time 0 then the probability of being in state B at time 1 is p.
[4] (a) If the system is started in state A at time 0, what is the probability it is in state A at time 2?
[2] (b) The transition matrix is stochastic. Is it regular? Why or why not?
7. Let T : R^6 → R^2 be a linear operator such that
T (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) =
x 1 + x 2 + x 3 3
x 4 + x 5 + x 6 3
[3] (a) What is [T ], the matrix of T?
[2] (b) What is the rank of [T ]?
[2] (c) Give a basis for the range of T.
[2] (d) What is the nullity of [T ]?
[4] (e) Give a basis for the kernel of T.
9. Let A =
. Let P =
√^1 2 −^ √^1 1 2 0 √ 2 √^1 2 0 0 0 1
. Note that^ P^ is an orthogonal matrix.
Let B = P AP −^1.
[1] (a) What is P −^1?
[2] (b) What are the eigenvalues λ 1 , λ 2 , λ 3 of B?
[3] (c) Give eigenvectors x 1 , x 2 , x 3 that corresponds to each eigenvalue of B in (b)?
[2] (d) Let v = (1, 1 , 1). Write v as a linear combination of eigenvectors of B.
[4] (e) What does Bkv converge to as k → ∞?
[4] (f) State the eigenvalues and corresponding eigenvectors of A.