Is b in the column space of A and is the system consistent?

• Benzoate
In summary, the conversation discusses determining if a given matrix and vector are consistent and if the vector is in the column space of the matrix. The matrix is a 2x2 matrix with specific values, and the vector is also given with specific values. The system is known to be consistent with infinitely many solutions, but it is unclear if b is in the column space of A. The question is whether b can be expressed as a linear combination of the columns of A.
Benzoate
1. Homework Statement [/b]

For each of the following choices of A and b, determine if b is the column space of A and state whether the system Ax=b is consistent

A is a 2 by 2 matrix , or A=(1,2,2,4) , 1 and 2 being on the first row and 2 and 4 on the second row. and b=[4,8] 4 being on the first row and 8 being on the second row . Ax=b

Homework Equations

3. The Attempt at a Solution

I know the system is consistent , because the system has infinitely many solutions. I haven't the first clue of how to determine if b is in the column space of A .

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Can b be expressed as a linear combination of the columns of A? If the system is consistent, well...

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1. What is a column space in a vector space?

A column space in a vector space is a subset of the vector space that is spanned by the columns of a given matrix. In other words, it is the set of all possible linear combinations of the columns of the matrix.

2. What is the dimension of a column space?

The dimension of a column space is the number of linearly independent columns in the matrix that spans the space. This is also known as the rank of the matrix.

3. How is a column space related to a basis?

A basis for a column space is a set of linearly independent vectors that span the column space. In other words, these vectors can be used to create any vector in the column space through linear combinations. The number of vectors in the basis is equal to the dimension of the column space.

4. Can a column space be empty?

No, a column space cannot be empty. Every vector space has at least one column space, which is the set of all possible linear combinations of the zero vector. This is known as the trivial column space.

5. How is a column space different from a row space?

A column space is the set of all linear combinations of the columns of a matrix, while a row space is the set of all linear combinations of the rows of the matrix. The number of vectors in the column space is equal to the number of columns in the matrix, while the number of vectors in the row space is equal to the number of rows in the matrix.

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