Linear Algebra Done Right: Solutions and Examples for Success

College
Algebra
Linear algebra

I am trying to interpolate a function defined over a three-dimensional real space:
$f : R^{3} \to R (x, y, z) \to f (x, y, z)$
Let assume I have $N_{1} N_{2} N_{3}$ points in the space which form my grid for this interpolation, and the multivariate series
$F (x, y, z) = \sum_{a = 0}^{N_{1} - 1} \sum_{b = 0}^{N_{2} - 1} \sum_{c = 0}^{N_{3} - 1} C_{a b c} x^{a} y^{b} z^{c}$
is the chosen interpolator. In order to find the coefficients I should form the following sets of equations:
$\sum_{a = 0}^{N_{1} - 1} \sum_{b = 0}^{N_{2} - 1} \sum_{c = 0}^{N_{3} - 1} C_{a b c} x_{i}^{a} y_{i}^{b} z_{i}^{c} = f_{i} = f (x_{i}, y_{i}, z_{i}), where, i = 1, \dots, N_{1} N_{2} N_{3},$
and then solve for the coefficients. However, for this I first need to write the above set of linear equations in the standard form
$A \vec{x} = \vec{b}$
wherein, A is the matrix of coefficients, $\vec{x}$ is the vector of unknowns, and $b = {f_{i}}_{1}^{N_{1} N_{2} N_{3}}$ is the known vector. For this to be done I would require to expand the multivariate power series in the form of a single variable series, that is,
$\sum_{a = 0}^{N_{1} - 1} \sum_{b = 0}^{N_{2} - 1} \sum_{c = 0}^{N_{3} - 1} C_{a b c} x_{i}^{a} y_{i}^{b} z_{i}^{c} = \sum_{m = 0}^{(N_{1} - 1) (N_{2} - 1) (N_{3} - 1)} C_{m} η_{i}^{m}$
wherein $η_{i} = η_{i} (x_{i}, y_{i}, z_{i})$ and probably $m = a b c$ . OF course the expansion needs to be nontrivial and useful.
Is it possible at all? Any suggestion to find the coefficients more practically?
Regards, owari
UPDATE.
Maybe it appears that the most natural way for solving this problem is comprised of the following steps:
1. first solve for the coefficients of
$\sum_{a = 0}^{N_{1} - 1} E_{a} (y, z) x_{i}^{a} = f (x_{i}, y, z), where, i = 1, \dots, N_{1},$
2. then solve for the coefficients of
$\sum_{b = 0}^{N_{2} - 1} D_{a b} (z) y_{i}^{b} = E_{a} (y_{i}, z), where, i = 1, \dots, N_{2}, and, a = 0, \dots, N_{1} - 1$
3. and finally solve for the coefficients of
$\sum_{c = 0}^{N_{2} - 1} C_{a b c} z_{i}^{c} = D_{a b} (z_{i}), where, i = 1, \dots, N_{3}, a = 0, \dots, N_{1} - 1, and, b = 0, \dots, N_{2} - 1$
which gives the coefficients we were looking for. However, this way, the first two steps will be working with function-type coefficients instead of numerical coefficients and that will prevent efficient usage of the available codes in numerical analysis. Solving for each set of coefficients at each step for different grid points would also inevitably increase the number of equations drastically, so any better suggestion to solve for this problem?

For any vectors u, v and w, show that the vectors u+v, u+w and v+w form a linearly dependent set.

Write in words how to read each of the following:
{x∈I ├|0<x<1┤}
{x∈R├|x≤0 or x≥1┤}

The symmetric difference of A and B, denoted by A B, is the set containing those elements in
either A or B, but not in both A and B.
a) Find the symmetric difference o f { 1, 3 , 5 } and { 1, 2, 3 } .
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A (B C) = (A B) C?

DAWIT BERIHUN2022-04-02

Let f be the function from R to R defined by f(x) = x2 •
Find
a) f-1 ({1}).
b) f-1 ({x \ 0 < x < I }).
c) f-1 ({x \ x > 4}).

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Find k $k$ such that the following matrix M $M$ is singular.
M=⎡⎣⎢−225+k−5933−4114⎤⎦⎥

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Let AB = Im, where A is an m×n matrix and B is an n×m matrix. If y ∼ MVN(μ,In) and BA is symmetric, find the distribution of yT BAy.

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describe the motion of the particle if normal component of acceleration is 0

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Determine which of the following transformations are linear transformations.

A. The transformation T defined by T(x1,x2,x3)=(1,x2,x3)T(x1,x2,x3)=(1,x2,x3)
B. The transformation T defined by T(x1,x2)=(2x1−3x2,x1+4,5x2)T(x1,x2)=(2x1−3x2,x1+4,5x2).
C. The transformation T defined by T(x1,x2)=(4x1−2x2,3|x2|)T(x1,x2)=(4x1−2x2,3|x2|).
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vv
A second hand car dealer has ten cars for sale. He decides to investigate the link between the
ages of the cars, x years and the mileage, y thousand miles. The data collected from the cars
are shown in the table below.
Age, x years
2 2.5 3 4 4.5 4.5 5 3 6 6.5
Mileage, y thousand miles
22 34 33 37 40 45 49 30 58 58
(a) Construct a well labeled scatter diagram in the graph paper provided to present the
information. (3 marks)
(b) Find Sxx and Sxy. (3 marks)
(c) Find the regression equation in the form y = α + βx. (4 marks)
(d) Fit the regression equation obtained in (c) to the scatter plot in (a) and label the line
appropriately. (2 marks)
(e) Give a practical interpretation of the slope β. (1 mark)
(f) Using your answer in (c), fifind the mileage for a 6 year old car.
A second hand car dealer has ten cars for sale. He decides to investigate the link between the
ages of the cars, x years and the mileage, y thousand miles. The data collected from the cars
are shown in the table below.
Age, x years
2 2.5 3 4 4.5 4.5 5 3 6 6.5
Mileage, y thousand miles
22 34 33 37 40 45 49 30 58 58
(a) Construct a well labeled scatter diagram in the graph paper provided to present the
information. (3 marks)
(b) Find Sxx and Sxy. (3 marks)
(c) Find the regression equation in the form y = α + βx. (4 marks)
(d) Fit the regression equation obtained in (c) to the scatter plot in (a) and label the line
appropriately. (2 marks)
(e) Give a practical interpretation of the slope β. (1 mark)
(f) Using your answer in (c), fifind the mileage for a 6 year old car.
A second hand car dealer has ten cars for sale. He decides to investigate the link between the
ages of the cars, x years and the mileage, y thousand miles. The data collected from the cars
are shown in the table below.
Age, x years
2 2.5 3 4 4.5 4.5 5 3 6 6.5
Mileage, y thousand miles
22 34 33 37 40 45 49 30 58 58
(a) Construct a well labeled scatter diagram in the graph paper provided to present the
information. (3 marks)
(b) Find Sxx and Sxy. (3 marks)
(c) Find the regression equation in the form y = α + βx. (4 marks)
(d) Fit the regression equation obtained in (c) to the scatter plot in (a) and label the line
appropriately. (2 marks)
(e) Give a practical interpretation of the slope β. (1 mark)
(f) Using your answer in (c), fifind the mileage for a 6 year old car

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x + y - z = -22x - y + 3z = 9x - 4y - 2z = 1find the solution

Finding detailed linear algebra problems and solutions has always been difficult because the textbooks would never provide anything that would be sufficient. Since it is used not only by engineering students but by anyone who has to work with specific calculations, we have provided you with a plethora of questions and answers in their original form. It will help you to see some logic as you are solving complex numbers and understand the basic concepts of linear Algebra in a clearer way. If you need additional help or would like to connect several solutions, compare more than one solution as you approach your task.

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