Math 341 Practice Final
1) Given the points A = (3, −2, 1), B = (4, 0, 1), C = (1, −2, 3), D = (3, 2, 3) :
i)
Find the vector form of the equation of the plane passing through A, B and C.
ii) Determine if the point D is on the plane, or not.
iii) Find the angle ∠ABD (i.e. the angle betweeen the line segment from B to A and the line segment from B to D)
in degrees.
2) Claire runs a catering business where she sells fruit salad.
For one event, she buys 20 pounds of oranges, 15 pounds of bananas and 15 pounds of apples, for total cost of $91.25.
For a second event, she buys 35 pounds of oranges, 20 pounds of bananas and 10 pounds of apples, for total cost of
$127.50.
For a third event, she buys 30 pounds of oranges, 30 pounds of bananas and 20 pounds of apples, for total cost of $142.50.
If the cost of the fruits was the same for all three events, determine the cost (per pound) of each of these fruits.
4
−3
−3, D = −3
−3
1
−0
−3 3 −2
−3 5
3
1 , E = −4 3 −3, and F = 0
1
1
1
4 −6 −3
4 −2
3
1 0 0
It is the case (I am telling you this) that CD = EF = 0 1 0.
0 1
0
1
i) Use this information to find all x satisfying ECx = 1
0
ii) Use this to calculate DC 3 D2 F 2 E.
iii) What is the rank of D? The rank of C?
iii) Use this to find (CF )−1
−7 0 9
4) Let A = 3 2 −3.
−6 0 8
i)
Determine all of the eigenvalues of A
ii) For each eigenvalue λ out of the ones you found, find a basis for Eλ .
iii) Is A diagonalizable? If so, find matrices P and diagonal matrix D such that A = P DP −1 .
5
3) Let C = −4
−5
−4
3
4
−4
−5
0
x
5) Let S be the subset of R3 defined by S = {y : |x| ≤ 5}
z
i)
What is the definition of a linear subspace of Rn ?
ii) Give two different vectors that are in S
iii) Is S a linear subspace of R3 ? If it is, give a proof why. If it is not, give a counterexample.
2
3
1
6) Given the vectors u1 = 2 , u2 = 1 ,u3 = −5 ,
1
2
2
a) Determine if these 3 vectors are linearly independent. If they are not, find a dependency relationship.
b) Find the angle between u1 and u2 .
7) Solve the system of equations
x + 2y = 2
y − 2z = 0
−4x − 6y − 5z = −9
Show all steps.
8) For each of the following vector valued functions, determine if the function is a linear transformation. If it is, give the
standard matrix of the transformation. If it isnot a linear transformation,
explain why not using a specific counterexample.
2x1 + x2
x1
i)
T : R2 → R3 given by T
= 3x1 − 4x2
x2
0
2
x
1
ii) T : R2 → R3 given by T
= 3x1 + 2x2
x2
6x1 − 2x2
9) Consider the statements below. For each statement, decide if it is true or false. If it is true, give an explanation (you do
not need to give a full proof), if it is false then give a counterexample.
i)
If A and B are n × n matrices, and rank(AB) < n, then it must be that rank(B) < n.
ii)
If det(A) 6= 0, then the rows of A are linearly independent.
iii)
If A is an n × n matrix and 0 is an eigenvalue of A, then the rank of A is less than n.
iv)
If A is n × n and col(A) is all of Rn , then det(A) = 0.
v)
If A is an m × n matrix, and b ∈ Rm is in the column space of A, then Ax = b is a consistent linear system.
3
4
2
10) Let A = 2 −1 3 .
−1 6 −4
i) Find a basis for row(A)
ii) Find a basis for col(A)
iii) Find a basis for null(A)
iv) Determine the rank and nullity of A.
11) i) Find the coefficients of the quadratic polynomial y = ax2 + bx + c that satisfies f (1) = 5, f (2) = 18, and f 0 (2) = 16.
ii) Find the coefficients of the linear polynomial y = mx + b that is the least-squares fit to the data given above.
3 2 7
2 0 y1
1
12) Find the vector y that is the least-squares solution to the overdetermined system
1 5 y2 = 6
−2 0
−3
b 0 2
13) Let A = 3 2 −1.
−2 0 1
i)
Find the value of b if det(A) = 20.
ii)
Find the value of b if A is not an invertible matrix.
14) Consider the following electrical component, which has two inputs v1 , v2 , three outputs v3 ,v4 , and v5 , and a ground
terminal.
v1#
v3#
v4#
v2#
v5#
When the inputs are set to v1 = 8V and v2 = 4V , the output voltages are v3 = 1.5V , v4 = 2.5V and v5 = 4V .
When the inputs are set to v1 = 4V and v2 = 4V , the output voltages are v3 = 1V , v4 = 1.5V and v5 = 2.5V .
Assume that this device is linear. The given information is sufficient to fully characterize the device.
i)
Determine what the output must be if v1 = 1V and v2 = 0V .
ii)
Determine what the output must be if v1 = 0V and v2 = 1V .
iii)
v3
v
Find a matrix M so that the outputs and inputs are related by v4 = M 1
v2
v5
0
