In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.
Let
F
{\displaystyle \mathbb {F} }
be a field. The column space of an m × n matrix with components from
F
{\displaystyle \mathbb {F} }
is a linear subspace of the m-space
F
m
{\displaystyle \mathbb {F} ^{m}}
. The dimension of the column space is called the rank of the matrix and is at most min(m, n). A definition for matrices over a ring
K
{\displaystyle \mathbb {K} }
is also possible.
The row space is defined similarly.
The row space and the column space of a matrix A are sometimes denoted as C(AT) and C(A) respectively.This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces
\begin{pmatrix}
2 & 4 & 6 \\
3 & 5 & 8 \\
1 & 2 & 3
\end{pmatrix}
Using the row operations, R2<-- R2-3R1 R3<-- R3-R1 we find the row echelon form of the matrix.
\begin{pmatrix}
1 & 2 & 3 \\
0 & -1 & -1 \\
0 & 0 & 0
\end{pmatrix}
Based on the definition of row space in the book Í am...
Summary:: x
Question:
Book's Answer:
My attempt:
The coordinate vectors of the matrices w.r.t to the standard basis of ## M_2(\mathbb{R}) ## are:
##
\lbrack A \rbrack = \begin{bmatrix}1\\2\\-3\\4\\0\\1 \end{bmatrix} , \lbrack B \rbrack = \begin{bmatrix}1\\3\\-4\\6\\5\\4 \end{bmatrix}...
I'm doing problems on finding row and column spaces. My textbook tells me to find the echelon form of the matrix, and then to identify the bases. My question is, can I reduce the matrix to reduced echelon form to get the bases? I have the same question about bases for the solution space.
Homework Statement
We're given some linear transformations, and asked what the null space, column space and row space of the matrix representations tell us
Homework EquationsThe Attempt at a Solution
I know what information the column space and null space contain, but what does the row space of...
Suppose, ##A## is an idempotent matrix, i.e, ##A^2=A##.
For idempotent matrix, the eigenvalues are ##1## and ##0##.
Here, the eigenspace corresponding to eigenvalue ##1## is the column space, and the eigenspace corresponding to eigenvalue ##0## is the null space.
But eigenspaces for distinct...
Say a subspace S of R^3 is spanned by a basis = <(-1,2,5),(3,0,3),(5,1,8)>
By putting these vectors into a matrix and reducing it to rref, a basis for the row space can be found as <(1,-2,-5),(0,1,3)>. Furthermore, the book goes on to say that this basis spans the subspace S.
Cool, not...
In my linear algebra class we previously studied how to find a basis and I had no issues with that. Now we are studying the basis of a row space and basis of a column space and I'm struggling to understand the methods being used in the textbook. The textbook uses different methods to find these...
Hi,
My textbooks say that when a solution, x, is found to Ax=b it has a particular solution, x_0, such that A*x_0=b which is then combined with other solutions from the null space, n_i, such that A*n_i=0.
However, when playing about with this I seem to have come across a problem.
for...
Hello,
Homework Statement
Could anyone please explain why the row space of a matrix mXn over R is a subspace of R^n, and not of R^m?
Homework Equations
The Attempt at a Solution
1.Construct a matrix whose null space consists of all linear combination of the vectors, v1={1;-1;3;2} and v2={2,0,-2,4} (v1,v2 are column vector).2.The equation x1+x2+x3=1 can be viewed as a linear system of one equation in three unknowns. Express its general solution as a particular solution...
"Ax=b; every vector b is exactly from one vector x (from row space of A)".. <more>?
Hi,
I m referring 'Linear Algebra and its applications by Gilbert Starng".
I read (ch.3.1)"Matrix transforms every vector from its row space to its column space". Or
if given Ax=b; every vector b is...
Homework Statement
How to verify that the nullspace is orthogonal to the row space of B?
I have inserted the screen-shot of the problem below:
http://i29.fastpic.ru/big/2011/0918/10/ca341692cc37b831143f5fe32351db10.jpg
Homework Equations
Nullspace and orthogonality.The Attempt at a Solution
I...
Homework Statement
Find the basis for the row space
The Attempt at a Solution
the given matrix is
0 1 2 1
2 1 0 2
0 2 1 1
So i reduced to row-echeleon form
2 1 0 2
0 1 2 1
0 0 3 1
so then rank = 3. My textbook states that the basis of the row space are the row vectors of leading ones...
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...
A =
1 2 -1 3
3 5 2 0
0 1 2 1
-1 0 -2 7
Problem: Find a basis for the row space of A consisting of vectors that are row vector of A.
My attempt:
I transpose the matrix A and put it into reduced row echelon form. It turns out that there are leading ones in every column...
If n*n matrix, can row space ever be equal to null space?
P.S.: this is NOT a homework question. It's a general question to get the concepts straight in my head.
Homework Statement
I have a question about an old exam and i have no idea where to start. The question says:
Find the vector in the row space A=[1 0 1 1 /new row 0 1 -1 1] 2x4 matrix nearest to the vector (1,1,-1,1)
I have absolutely no clue how to start.
Homework Equations
The...
Homework Statement
What are
the basis for the row space and null space for the following matrix? Find the dimension of RS, dim of NS.
[1 -2 4 1]
[3 1 -3 -1]
[5 -3 5 1]
Homework Equations
dim RS + dim NS = # of columns
The Attempt at a Solution
I reduced the matrix into...
In my homework problem, I am supposed to find out whether an element belongs to the rowspace of a matrix. So, what I did is to determine the (row)basis of the matrix, dimension of it being one row less of the rows of the original matrix. So, instead of the linearly-dependent row I put the...