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Construct a 3 x 3 matrix, not in echelon form, whose columns span R³. Show that the matrix you construct has the desired property.

Construct a 3 x 3 matrix, not in echelon form, whose columns span R³. Show that the matrix you construct has the desired property.

C++ for Engineers and Scientists
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter13: Structures
Section: Chapter Questions
Problem 4PP
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2 1 3 X1 X2 -H 0 27. Let 9₁, 92, 93, and v represent vectors in R5, and let x₁, x2, and x3 denote scalars. Write the following vector equation as a matrix equation. Identify any symbols you choose to use. X19₁ + x₂9₂ + X3q3 = V 28. Rewrite the (numerical) matrix equation below in symbolic form as a vector equation, using symbols V₁, V2,... for the vectors and C₁, C2, ... for scalars. Define what each symbol represents, using the data given in the matrix equation. SG 71 -3 5 -4 9 7 581 -2 -4 -37 2 4 -1 2 8 -[-] 29. Construct a 3 x 3 matrix, not in echelon form, whose columns span R³. Show that the matrix you construct has the desired property. 30. Construct a 3 x 3 matrix, not in echelon form, whose columns do not span R³. Show that the matrix you construct has the desired property. 31. Let A be a 3 x 2 matrix. Explain why the equation Ax = b cannot be consistent for all b in R³. Generalize your Mastering Linear Algebra Concepts: Span 1-18 1. The matrix equation [M] In span R 37. 39. be СО 40. 41. [ 1 42. L WE SOLUTIONS TO PRACTICE I 1
Transcribed Image Text:2 1 3 X1 X2 -H 0 27. Let 9₁, 92, 93, and v represent vectors in R5, and let x₁, x2, and x3 denote scalars. Write the following vector equation as a matrix equation. Identify any symbols you choose to use. X19₁ + x₂9₂ + X3q3 = V 28. Rewrite the (numerical) matrix equation below in symbolic form as a vector equation, using symbols V₁, V2,... for the vectors and C₁, C2, ... for scalars. Define what each symbol represents, using the data given in the matrix equation. SG 71 -3 5 -4 9 7 581 -2 -4 -37 2 4 -1 2 8 -[-] 29. Construct a 3 x 3 matrix, not in echelon form, whose columns span R³. Show that the matrix you construct has the desired property. 30. Construct a 3 x 3 matrix, not in echelon form, whose columns do not span R³. Show that the matrix you construct has the desired property. 31. Let A be a 3 x 2 matrix. Explain why the equation Ax = b cannot be consistent for all b in R³. Generalize your Mastering Linear Algebra Concepts: Span 1-18 1. The matrix equation [M] In span R 37. 39. be СО 40. 41. [ 1 42. L WE SOLUTIONS TO PRACTICE I 1
Expert Solution
Step 1

Here's one example of a 3x3 matrix whose span is R^3:

1  2  3
4  5  6
7  8  9

 

To show that this matrix has the desired property, we need to demonstrate that the set of all linear combinations of its columns forms the entire 3-dimensional space. This can be done by showing that the columns are linearly independent and that they span the space.

 

Linear independence: The columns of the matrix are linearly independent, meaning that no column can be expressed as a linear combination of the others. For example, the first column (1, 4, 7) cannot be expressed as a combination of the second and third columns (2, 5, 8) and (3, 6, 9).

 

 
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