1. Give an example where a combination of three nonzero vectors in Rª is the zero vector. Then writeyour example in the form Ax = 0. What are the dimensions of matrix A and x and 0 vector?2. Suppose a combination of the columns of A equals a different combination of those columns.Write it as Ax = Ay. Find two combinations of the columns of A that equal the zero vector (in matrixlanguage, find two solutions to Az = 0 vector in terms of x and y).3. Suppose A is the 3 by 3 matrix ones(3, 3) of all ones. Find two independent vectors x and y thatsolve Ax = 0 and Ay = 0. Why don't I ask for a third independent vector with Az = 0 ?4. The linear combinations of v = (1, 1, 0) and w = (0, 1, 1) fill a plane in R3 .(a) Find a vector z that is perpendicular to v and w(b) Find a vector u that is not on the plane. Justify your answer5. If three corners of a parallelogram are (1, 1), (4, 2), and (1, 3), what are all three of the possiblefourth corners? Draw two of them.Go through the slides on orthogonality and write matlab code6. Write a matlab code for creating 10x10 real Fourier basis matrix in a loop.7. Write a matlab code for creating 11x11 real Fourier basis matrix in a loop.8. Write a matlab code for creating 10x10 DCT matrix in a loop.Write a matlab code for creating 10x10 DST matrix in a loop.10. Write a Matlab code for creating a 5x5 random integer matrix (matrix with integer elements)write a code for getting diagonal sum and antidiagonal sum.(1 2 347Let A =9 10 11 12(13 14 16 17.main_diag_sum =1+6+11+17,anti _ diag _ sum = 4 +7+10+1311. Write a Matlab code for creating multiplication table for number from 1 to 10 using outerproduct of vectors.(multipliers up to 16. Example for number 9, compute 1x9, 2x9,...to 16x9)12. Use outer_product for creating non_normalized bases in problem 6,7,8,9 in one or two lines ofcode

question 1 let x1=2,0,0,0x2=0,2,0,0x3=2,2,0,0 where x1,x2,x3∈ℝ4 and let α1=1 , α2=1 , α3=-1

Answered: 1. Give an example where a combination…

1. Give an example where a combination of three nonzero vectors in R* is the zero vector. Then write your example in the form Ax = 0. What are the dimensions of matrix A and x and 0 vector?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

1. Give an example where a combination of three nonzero vectors in Rª is the zero vector. Then write
your example in the form Ax = 0. What are the dimensions of matrix A and x and 0 vector?
2. Suppose a combination of the columns of A equals a different combination of those columns.
Write it as Ax = Ay. Find two combinations of the columns of A that equal the zero vector (in matrix
language, find two solutions to Az = 0 vector in terms of x and y).
3. Suppose A is the 3 by 3 matrix ones(3, 3) of all ones. Find two independent vectors x and y that
solve Ax = 0 and Ay = 0. Why don't I ask for a third independent vector with Az = 0 ?
4. The linear combinations of v = (1, 1, 0) and w = (0, 1, 1) fill a plane in R3 .
(a) Find a vector z that is perpendicular to v and w
(b) Find a vector u that is not on the plane. Justify your answer
5. If three corners of a parallelogram are (1, 1), (4, 2), and (1, 3), what are all three of the possible
fourth corners? Draw two of them.
Go through the slides on orthogonality and write matlab code
6. Write a matlab code for creating 10x10 real Fourier basis matrix in a loop.
7. Write a matlab code for creating 11x11 real Fourier basis matrix in a loop.
8. Write a matlab code for creating 10x10 DCT matrix in a loop.
Write a matlab code for creating 10x10 DST matrix in a loop.
10. Write a Matlab code for creating a 5x5 random integer matrix (matrix with integer elements)
write a code for getting diagonal sum and antidiagonal sum.
(1 2 3
4
7
Let A =
9 10 11 12
(13 14 16 17.
main_diag_sum =1+6+11+17,
anti _ diag _ sum = 4 +7+10+13
11. Write a Matlab code for creating multiplication table for number from 1 to 10 using outer
product of vectors.
(multipliers up to 16. Example for number 9, compute 1x9, 2x9,...to 16x9)
12. Use outer_product for creating non_normalized bases in problem 6,7,8,9 in one or two lines of
code

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