Linear Algebra: 4 Fundamental Subspaces

In summary, the problem asks for the bases of the four fundamental subspaces without directly computing matrix A. The row space's basis can be determined from the three rows of U, similar to an example in the book. The column space's basis can be found from the three columns of L. To find the nullspace, U can be put into row reduced echelon form and the nullspace basis can be solved for. There is no left nullspace since there is no zero row.
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ChEJosh
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Homework Statement



Without computing A, find the bases for the 4 fundamental subspaces.

[1 0 0][1 2 3 4]
[6 1 0][0 1 2 3]=A=LU
[9 8 1][0 0 1 2]

Homework Equations



N/A

The Attempt at a Solution



There was an "example" in the book. It just showed the answers.
It was:

[1 0 0][1 3 0 5]
[2 1 0][0 0 1 6]=A
[5 0 1][0 0 0 0]
Where
Row Space: Basis (1,3,0,5) and (0,0,1,6)
Column Space: Basis (1,2,5) and (0,1,0)

In the problem we have to do, I take it that the row space's basis is just the 3 rows of U similar the the example. But, I'm unsure of the column space. Is it the 3 columns of L?

And, also for the nullspace, I put U in its row reduced echelon form, and solved for the nullspace as one would normally do. Is this correct?

[1 2 3 4]
[0 1 2 3] -->
[0 0 1 2]

[1 0 0 0]
[0 1 0 -1]
[0 0 1 2]
So, the nullspace basis is (0,1, -2, 1)

And, since there isn't a zero row, there also isn't a left nullspace, correct?

Sorry if that's confusing. And, thank you in advance!
 
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1. What is Linear Algebra and why is it important?

Linear Algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It is important because it provides the foundation for many other fields in mathematics and has numerous applications in physics, engineering, economics, and computer science.

2. What are the 4 fundamental subspaces in Linear Algebra?

The 4 fundamental subspaces in Linear Algebra are the column space, null space, row space, and left null space. These subspaces represent all possible combinations of the columns and rows in a matrix and provide important information about its structure and properties.

3. How are the 4 fundamental subspaces related to each other?

The column space and the row space are orthogonal complements of each other, meaning that they are perpendicular in a vector space. The null space and left null space are also orthogonal complements of each other. Additionally, the column space and null space are complementary, as are the row space and left null space.

4. What are the uses of the 4 fundamental subspaces in real-world applications?

The column space and row space are used to solve systems of linear equations and to determine the rank and dimension of a matrix. The null space and left null space are used in data compression and image processing, as well as in solving optimization problems.

5. How can one find the basis for each of the 4 fundamental subspaces?

The basis for the column space and row space can be found by performing row operations on a matrix and identifying the pivot columns. The basis for the null space and left null space can be found by solving the homogeneous system of equations using Gaussian elimination and identifying the free variables.

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