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Linear Algebra: Vectors and Least Squares Problems

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Please see the attached document for formatted version of the problem.

1. Consider the set of vectors:

V1 = [6, 0, 2]. V2 = [2, -1, 1], V3 = [-4, -1, -1]

Do they form a basis in R^2? Give reasons.

What is the dimension of space spanned by these vectors?

2. Find the orthogonal projection of the vector u = (-2, 2, 3) onto the subspace spanned by vectors v1 = (3, -1, 0) and v2 = (1, -2, 1).

3. Given the vectors (4, 2, 1), (2, -1, 1) and (2, 3, 0)
(a) Determine whether these vectors are linearly independent or dependent.
(b) What is the dimension of the space spanned by these vectors?

4. Find the least squares solution to the equation:
[1, 0, 1; -1, 1, 2]x = [0, 7, -1].

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Solution Summary

This posting explains how to find whether vectors are linearly independent or dependent and they form a basis or not. It also explains how to find the dimension of the space spanned by given vectors. It also includes step by step explanation to one question on finding least squares solution to a matrix equation.

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Please find solutions attached herewith.

See below for a condensed version of the solution. The attachment includes equations and detailed workings.

1/. Solution:

The vectors are linearly dependent since the determinant of the matrix is zero.
Since vectors are linearly dependent they can't form a basis in R^3.

multiply the 1st row by 1/6

add -2 times ...

Solution provided by:
Sandhya Sharma, MSc
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  • BSc, Meerut University
  • MSc, Meerut University
  • MPhil, Institute of Advanced Studies
  • MSc, AIT
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