Columns of Matrix Form a Basis?

In summary, the conversation discusses whether the columns of a matrix formed by the squared basis vectors of a given basis form a basis themselves. The speaker has found this to be true in their examples, but is unsure of how to prove that the columns are also linearly independent.
  • #1
puhsyers
1
0
Hi everyone,

This not a homework question. I'm reviewing some linear algebra and I found this on a worksheet. I just need a hint on how to approach this problem.

Let β=[itex]\{ v_1,v_2,...,v_n\}[/itex] be a basis for [itex]R^n [/itex]. Let M be the matrix whose columns are the basis vectors in β. Do the columns of [itex] M^2 [/itex] form a basis?

I've played around with some examples and it seems to be true. I know that M has full rank and hence invertible, therefore, [itex] M^2 [/itex] is also invertible and has full rank. But this only means the columns of [itex] M^2 [/itex] spans the vector space. How do I show the columns are also linearly independent?
 
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  • #2
If the columns are linearly dependent, then it doesn't have full rank. My definition of rank is the dimension of the space spanned by the columns.
 

1. What is the definition of "Columns of Matrix Form a Basis"?

The columns of a matrix form a basis when they are linearly independent and span the entire column space of the matrix.

2. How do you determine if the columns of a matrix form a basis?

To determine if the columns of a matrix form a basis, you can perform row operations on the matrix to reduce it to its reduced row echelon form (RREF). If the RREF has a pivot in every column, then the columns of the original matrix form a basis.

3. What is the importance of the columns of a matrix forming a basis?

The columns of a matrix forming a basis is important because it allows us to express any vector in the column space of the matrix as a linear combination of the basis vectors. This makes it easier to solve linear systems of equations and perform other operations on the matrix.

4. Can a matrix have more than one basis?

Yes, a matrix can have multiple bases as long as the columns of the matrix are linearly independent and span the entire column space. However, there is always a unique basis that is considered the standard basis for a given matrix.

5. How do you find the basis for a matrix?

To find the basis for a matrix, you can perform row operations to reduce the matrix to RREF. The columns with pivots in the RREF form the basis for the matrix. Alternatively, you can also use the Gram-Schmidt process to find an orthonormal basis for the matrix.

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