# Column, Solution and Row Spaces

• jeffreylze
In summary: But how do you check if a vector is in the column or row space of A?The vectors form a basis. Right?Yes, the vectors form a basis for the column space.
jeffreylze

## Homework Statement

A = [ 1 0 1
-1 1 -3
2 1 0]

what is the column space of A?

What is the row space of A

What is the solution space of A?

Also, if i were given a multiple choice, how do I check if a vector is in either the column, row or solution space of A ?

## The Attempt at a Solution

From my understanding, the column space is just V1 = (1,-1,2) V2 = (0,1,1) V3 = (1,-3,0) and the row space of A is w1 = (1,0,1) , w2 = (-1,1,-3) and w3 = (2,1,0) . For solution space, i have completely no idea.

The solution space is the set of column matrices X of dimension 3 by 1 such that AX=0.

The column space is v1, v2, and v3. But are v1, v2, and v3 linearly independent? You might want to find the basis for the column space.

But then again, how do I check if a certain vector is in the column or row space of A?

The vectors form a basis. Right?
So if there exists a linear combination of the vectors that equals another 'certain vector' then the 'certain vector' is in the column or row space.

Let's see c1*v1 + c2*v2 + c3*v3 = u

Can we solve for c1, c2, and c3? If so, then u is in the space.

This looks like a job for matrices to solve. See if u=(1 1 1) is in the column space.

Actually only V1 and V2 forms a basis, as v3 can be written as a linear combination of V1 and V2. C1*(1,-1,2) + C2*(0,1,1) = (1,1,1) . I formed a matrix A with (1,-1,2) and (0,1,1) in columns and then put (1,1,1) into the matrix in augmented form. But the system is inconsistent, no solutions. So u is not in the column space?

Right on both accounts. Despite my typepo, leaving out the word 'not', I think you are seeing how all of this fits together. Well done.

fantastic, at least now I know I am making some progress. =D

hokie1 said:
The column space is v1, v2, and v3. But are v1, v2, and v3 linearly independent? You might want to find the basis for the column space.
That's not the column space. It's the subspace of R^3 (for this problem) that is spanned by v1, v2, and v3. In other words, it's the set of vectors in R^3 that are linear combinations of v1, v2, and v3.

## 1. What is a column space?

A column space is the set of all possible linear combinations of the columns of a matrix. It represents the entire range of possible outputs that can be generated using the given set of column vectors.

## 2. How is the column space related to the rank of a matrix?

The column space of a matrix is directly related to its rank. The rank of a matrix is equal to the number of linearly independent columns in the matrix, which is also equal to the dimension of the column space.

## 3. What is a solution space?

A solution space is the set of all possible solutions to a system of linear equations. It represents the entire range of possible solutions that satisfy the given set of equations.

## 4. How is the solution space related to the null space of a matrix?

The solution space and the null space of a matrix are closely related. The null space represents the set of all solutions that have a zero output, while the solution space represents the set of all possible solutions. In other words, the solution space is the null space plus any particular solution to the system of equations.

## 5. What is a row space?

A row space is the set of all possible linear combinations of the rows of a matrix. It is essentially the transpose of the column space and represents the entire range of possible outputs that can be generated using the given set of row vectors.

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