How do i find the basis of subspace U

In summary, the conversation is about finding the basis of a subspace U, using elementary row operations to reduce a matrix, and identifying an error in the reduction process. It is suggested to check for a simple algebra mistake and clarify which subspace is being referred to.
  • #1
subopolois
86
0

Homework Statement


In this case find the basis of subspace U
1 2 3 4
5 6 7 8
-6 -8 -10 12

Homework Equations


elementary row operations


The Attempt at a Solution


alright, so i know i have to reduce the matrix and i have done so
1 2 3 4
0 1 2 3
0 0 0 1
now the answer i get is
[1 5 -6]^T [2 6 -8]^T [4 8 12]^T
but the answer in the back of my textbook is
[1 5 -6]^T [2 6 8]^T
what have i done wrong, it seems they have one less column, is it my reduction of the matrix?
 
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  • #2
You just had an error in reducing your matrix somewhere. Check over your work, because the matrix should have been
1 0 -1 -2
0 1 2 3
0 0 0 0
it was probably a simple algebra mistake.
 
  • #3
What "subspace U" are you talking about? You give an array of numbers which you refer to as a matrix. I can think of 4 different subspaces that might be meant here:

The row space of the matrix.
The column space of the matrix.
The null space (kernel) of the matrix.
The image of the matrix.

Since you are row-reducing the matrix, you probably mean the column space but it would be a good idea to say so!
 

FAQ: How do i find the basis of subspace U

1. What is a subspace?

A subspace is a subset of a vector space that contains all possible linear combinations of its vectors. In other words, it is closed under addition and scalar multiplication.

2. How do I determine if a set of vectors forms a basis for a subspace?

To determine if a set of vectors forms a basis for a subspace, you can follow these steps:

  • Check if the vectors are linearly independent, meaning that none of the vectors can be written as a linear combination of the others.
  • Check if the vectors span the entire subspace, meaning that every vector in the subspace can be written as a linear combination of the given vectors.
  • If both conditions are met, then the set of vectors forms a basis for the subspace.

3. How do I find the basis for a subspace using the reduced row echelon form?

To find the basis for a subspace using the reduced row echelon form, you can follow these steps:

  • Create a matrix with the given vectors as its columns.
  • Perform row operations to reduce the matrix to its reduced row echelon form.
  • The non-zero rows in the reduced matrix correspond to the basis vectors for the subspace.

4. Can a subspace have more than one basis?

Yes, a subspace can have more than one basis. This is because a basis is not unique and there can be multiple sets of linearly independent vectors that span the same subspace.

5. How can I use the Gram-Schmidt process to find a basis for a subspace?

The Gram-Schmidt process is a method for finding an orthogonal basis for a subspace. To use this process, you can follow these steps:

  • Start with a set of linearly independent vectors that span the subspace.
  • Find the orthogonal projection of each vector onto the subspace spanned by the previous vectors.
  • Subtract the projections from the original vectors to get a set of orthogonal vectors.
  • Normalize the orthogonal vectors to create an orthonormal basis for the subspace.

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