# Proving row space column space

• nhrock3
In summary: This means that the space spanned by the columns of A^T is contained within the space spanned by the columns of A.Similarly, if we consider the kth row of A, we know that this row is a linear combination of the columns of A, given by the components of (B^T)_k. This means that the space spanned by the rows of A is contained within the space spanned by the columns of A.Therefore, both spaces are contained within each other, and since they have the same dimensions (n), they must be equal. In other words, the space spanned by the rows of A is equal to the space spanned by the columns of A. In summary, by showing that the
nhrock3
A , B are nXn matrices
and
AB=(A)^t
t-is transpose
prove that the space spanned by A's row equals the space spanned by A's columns
i know that there dimentions are equals
so in order to prove equality i need to prove that one is a part of the other
how to do it?

each column i of (AB)_i=A*B_i
i was told by my proff that that column i of AB is a member from the span of the columns of A

but i don't get this result
suppose the member of B_i column is (c1,c2,..,cn)
so the multiplication of A by the B_i column
we get then the first member is dot product from the first row with (c1,c2,..,cn)
i can't see how its a variation from the A columns?

Last edited:
yeah i think you're on the right track,
A.B = A^T

now consider a the kth column B, the vector B_k, which when multiplied with A yields the kth column of A_T, (A^T)_k
A.B_k = (A^T)_k

so the kth column of A^T is a linear combination of the columns of A, given by the components of B_k.

## 1. What is the difference between row space and column space?

The row space of a matrix is the vector space spanned by the rows of the matrix, while the column space is the vector space spanned by the columns. In other words, the row space contains all linear combinations of the rows, while the column space contains all linear combinations of the columns.

## 2. How do you prove that two matrices have the same row space?

To prove that two matrices have the same row space, you can show that one matrix can be transformed into the other through elementary row operations. This means that the two matrices have the same rows, in a different order or with some rows multiplied by a scalar.

## 3. What does it mean for a matrix to have full row rank?

A matrix has full row rank if its row space spans the entire vector space in which it is defined. This means that every possible vector in the space can be written as a linear combination of the rows of the matrix.

## 4. How do you prove that a matrix has full column rank?

To prove that a matrix has full column rank, you can show that the columns are linearly independent, meaning that no column can be written as a linear combination of the other columns. This can be done by finding the determinant of the matrix, which must be non-zero for the columns to be linearly independent.

## 5. Why is it important to prove row space and column space properties?

Proving row space and column space properties is important in understanding the structure and properties of a matrix. This information can be used in solving systems of linear equations, finding the rank of a matrix, and determining the existence and uniqueness of solutions. It also helps in understanding the geometry of a matrix and its applications in various fields such as computer science, physics, and engineering.

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