# What is Column space: Definition and 46 Discussions

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

R

n

{\displaystyle \mathbb {R} ^{n}}
and

R

m

{\displaystyle \mathbb {R} ^{m}}
respectively.

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1. ### I Proof of Column Extraction Theorem for Finding a Basis for Col(A)

Theorem: The columns of A which correspond to leading ones in the reduced row echelon form of A form a basis for Col(A). Moreover, dimCol(A)=rank(A).
2. ### Linear independence of Coordinate vectors as columns & rows

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}...
3. ### B Finding Bases for Row and Column Spaces

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.
4. ### Does the Null Space of a 2x3 Matrix Determine its Column Space?

Homework Statement Let ##A## be a 2x3 matrix. If Nul(##A##) is a line through the origin in ℝ3, then Col(##A##) = ℝ2. Explain why. Hint: Think about the number of pivots in ##A##. Homework EquationsThe Attempt at a Solution So, Nul(##A##) is the set of all solutions to the equation ##Ax=0##...
5. ### I What is the Basis for the Null Space in Matrix A?

Hello there. I'm currently trying to come to terms with the aforementioned topics. As I am self studying, a full understanding of these concepts escapes me. There's something I'm not grasping here and I would like to discuss these to clear away the clouds. As I understand it, a basis for some...
6. ### I Are the columns space and row space same for idempotent matrix?

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...
7. ### MHB How to find a non-zero vector in the column space of M

Let the matrix $M = \begin{bmatrix}-12&-12&16&-15\\-6&-8&-8&-10\\0&20&0&25\end{bmatrix}$ Find a non zero vector in the column space of $M$ Is it not true that $\begin{bmatrix}-12\\-8\\20\end{bmatrix}$ is a non zero vector in the column space of $M$ ? For some reason it keeps telling me "that...
8. ### Column space and nullspace relationship?

I have just been studying Nullspaces... I want to make the following summary, will it be correct? C(A) is all possible linear combinations of the pivot columns of A. N(A) is all possible linear combinations of the free columns of A (if any exist). edit: I have a feeling these are...
9. ### MHB Relation between null and column space

Is there a relationship between 1 and 2. If so, is it 1 implies 2, 2 implies 1, or if and only if. 1) $\operatorname{null}A=\operatorname{null}B$ 2) $\operatorname{col}\operatorname{rref}A=\operatorname{col}\operatorname{rref }B$
10. ### How does L from LU give us column space?

I see that the pivots in columns 1 and 2 help us decide which columns to take. But why does the L matrix of this B = LU let just to read off the column space? 2:18
11. ### Question about row space basis and Column space basis

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...
12. ### Proving vectors are in the column space

How would you prove that adding two vectors in the column space would result in another vector in the column space? I know this is maybe the most basic property of vectors and subspaces, and that the very definition of the column space says it's spanned by vectors in the column space. Is there...
13. ### Why is Understanding Column Space and Null Space Important in Linear Algebra?

Why it is important to know about Column space and Null spaces in Linear Algebra?
14. ### MHB Understanding the Linear Independence of Columns in a 3x5 Matrix

i thought if A is 3x5, the columns of A must be linearly dependent, since the rank is at most 3, and the rank is the number of linearly independent columns in A. but there are 5 columns in A, so the columns of A must be linearly dependent :/
15. ### Linear Algebra - Basis of column space

Homework Statement Let A be the matrix A = 1 −3 −1 2 0 1 −4 1 1 −4 5 1 2 −5 −6 5 (a) Find basis of the column space. Find the coordinates of the dependent columns relative to this basis. (b) What is the rank of A? (c) Use the calculations in part (a) to...
16. ### Linear Algebra: Basis vs basis of row space vs basis of column space

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...
17. ### Column Space of Matrix A and ref(A)

Homework Statement Given a matrix A. So I can reduce A to ref(A). Let's say in ref(A), the columns that contain leading ones are column 1, 3, and 5. True or false: (a) Columns 1, 3, and 5 from ref(A) form the column space of ref(A). (b) The corresponding column 1, 3, and 5 from the original...
18. ### MHB Row Space, Column Space and Null Space

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...
19. ### Should be easy question on column space of a matrix

Homework Statement So the actual problem "Find the value of a for which the following system of linear equations has a solution" 2x + 4y + z = a -4x -7y + 0 = 1 0 -1y -2z = 1 Homework Equations The Attempt at a Solution I thought one approach was to find a basis for the...
20. ### Relationship between eigenspace and column space

Is it true that if an n by n matrix A has n-linearly independent eigenvectors, then it must also be invertible because these n-eigenvectors span n-space. But does this reasoning work the other way around: that is if A is invertible, does that imply n-linearly independent eigenvectors can be...
21. ### Relationship between column space of a matrix and rref of matrix

Hello, Does the column space of a matrix A always equal the column space of the rref(A)? i.e. are the solution sets to Ax=b, or even Ax=0 the same for A and rref(A)? When doing some examples of matrices that had some linearly independent columns it seemed the Span was preserved by row...
22. ### Nullspace, Column Space, and solution of system given only rref(A)

Homework Statement Suppose a 3 x 5 matrix A has row-reduced echelon form: [[1 2 0 0 5] [0 0 1 0 4] [0 0 0 1 3]] a. Describe NS(A) b. Describe CS(A) c. Suppose . [[2] . [3] [[-2] A [5] = [4] = b . [1] [3]] . [9]] To be clear, that's the original matrix A times the...
23. ### Column Space Problem: Obtain CS(B)

Homework Statement Obtain the column space of the following matrix B = 2 -3 -1 2 -3 -1 -3 3 2 Homework Equations Linear independence test c1V1 + c2V2 + ... + cnVn = 0 c1=c2=...cn=0 The Attempt at a...
24. ### Column Space of A'*A: Subset of A'?

Let A be an n x p matrix with real entries and A' be its transpose. Is the column space of A'*A the same as the column space of A'. Obviously, the column space of A'*A is a subset of the column space of A' but can I show the other way? Thanks!
25. ### Proving row space column space

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...
26. ### Prove the column space of AB is contained in the column space of matrix A

lets assume the matrix multiplication AB exists, how would i prove that the column space of AB is contained in the column space of matrix A? i know there is a theorem that says something like: "a system of linear equations Ax=b has solutions if and only if b is in the column space of A" Am...
27. ### Column space of positive semidefinite matrix

how to prove that R(A)=\text{sum of} N(A-\lambda I)? \lambda is nonzero eignevalues of A
28. ### Properties Of Matrices with the same Column Space

Homework Statement Suppose that A and B are 5 x 5 matrices with the same Column Space (image). (a) Must they have the same columns? (b) Must they have the same rank? (c) Must they have kernels of the same dimension? (d) Must they have the same kernel? (e) If A is invertible, must B be...
29. ### Column Space Basis: Why Does Row Reduction Work?

I am a bit puzzled by the following. You know how they teach you that in order to find column space you just need to row reduce the matrix, look at the columns with leading 1's in them and then just read off those columns from the original matrix? Well, why does that actually work? I'm trying to...
30. ### Howto define the Column space of nxn matrix

Homework Statement I thought that if you have a square matrix then the column space is the set of all linear independent vectors which can be written as a linear combinations of the others? Which inturn is the same as range of the Matrix? Am I wrong?
31. ### [Linear Algebra] Nullspace equals Column space

Homework Statement Why does no 3 by 3 matrix have a nullspace that equals its column space? Homework Equations NA The Attempt at a Solution A = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \] C(A) = \begin{bmatrix} 1 \\ 0 \\ 0...
32. ### Column Space and Pivot Columns in Reduced Matrices

To find the column space of a matrix, you reduce the matrix and those columns that contains leading variables(pivot columns), refers to the columns in the original matrix who span the columnspace of the matrix. But does the pivotcolumns in the reduced matrix also span the column space of the...
33. ### Finding the Column Space of Matrix A

Homework Statement We have a matrix A which row-reduces to: A = \left[\begin{array}{ccccc} 1&2&0&0\\ 0&0&1&0\\0&0&0&1 \end{array}\right] I'm asked to find the column space of A. Homework Equations The Attempt at a Solution I'm not sure what to write down for this...
34. ### What Does It Mean When Col A Is a Subspace of the Null Space of A?

I am just wondering what is meant when someone says the Col A is a subspace of null Space of A. What can you infer from that? Also what is a null space of A(transpose)A How do they relate to A? Are there theorems about this that I can look up?
35. ### Dimension of row/ column space

Homework Statement In the following exercises verify that the row rank is equal to the column rank by explicitly finding the dimensions of the row space and the column space of the given matrix. A = [1 2 1 ; 2 1 -1] Homework Equations The Attempt at a Solution All i can...
36. ### Does [2 15]T Lie in the Column Space of A?

Homework Statement Does b = [ 2 15 ]T lie in the column of the matrix A [1 -3] [2 5] Homework Equations a basis of CS(U) forms a basis for the corresponding columns for CS(A) The Attempt at a Solution I reduced the given matrix into row echelon form and determined its column...
37. ### Basis for row and column space

Homework Statement Can anyone help me figure out basis for RS(A) and basis for CS (A) along with their dimension? I mean dim CS(A) and dim RS(A) where A is [1 -2 4 1] [0 7 -15 -4] [0 0 0 0] Homework Equations The Attempt at a Solution are all non zero rows the basis for...
38. ### Understanding row and column space

Understading row and column space Homework Statement I am having hard time trying to understand row and column space. Can anyone simplify the meanings of them so that i can visualize them well. Homework Equations dimension of row space = rank ? How? Why? The Attempt at a Solution...
39. ### How Do You Determine the Column Space and Kernel of a Matrix?

Homework Statement If col (A) is column space of A and ker(A) null space of A with ker(A) = {Ax = 0} and ker(A') = {A'y = 0} Homework Equations Consider the (3x2) matrix : A = [1,2 ; 3,4 ; 5,6] (matlab syntax) Show that col(A) = c1 * [1,0,-1]' + c2 * [0,1,2]' The Attempt...
40. ### Solve Column Space, Matrix Problem with (x,y,z,w)^T

[SOLVED] Column space, matrix Homework Statement I have a linear transformation f from R^4 -> R^4 given by a matrix. I have to find the range of f(R^4) which containts the vector (x,y,z,w)^T. The Attempt at a Solution I know that the range of f is the column space, how do I make sure that...
41. ### Proving Theorem: Column Space of Matrix A is a Subspace of R^m

How would I prove this theorem: "The column space of an m x n matrix A is a subspace of R^m" by using this definition: A subspace of a vector space V is a subset H of V that has three properties: a) the zero vector of V is in H. b) H is closed under vector addition. c) H is closed under...
42. ### Rank, Dimension, Subsapce, Column Space

1) True or False? If true, prove it. If false, prove that it is false or give a counterexample. 1a) If A is m x n, then A and (A^T)(A) have the same rank. 1b) Let A be m x n and X E R^n. If X E null [(A^T)(A)], then AX is in both col(A) and null(A^T). [I believe it's true that AX is in...
43. ### Finding Orthonormal Set q1, q2, q3 for Column Space of A

I just need a hint. Problem: find an orthonormal set q1, q2, q3 for which q1, q2 span the column space of A, where A = [1 1] [2 -1] [-2 4] of course I should apply the Gram-Schmidt method, but the problem is that the column vectors are not independent and Gram-Schmidt starts with...
44. ### Row and Column space questions.

Hey, I was looking for help on these questions dealing with row and column spaces... 1. Prove that the linear system Ax = b is consistent IFF the rank of (A|b) equals the rank of A. 2. Show that if A and B are nxn matrices, and N(A-B) = R^n, then A = B The first one I can't get much...
45. ### Confused about Column Space? Let Us Help!

so i tried looking it up on various sources including wikipedia, and i am still confused about column space actually is. maybe it would help if one of you explained it to me?
46. ### Nullspace and Column Space Question

Nullspace and Orthogonal Complement Quick question: is the nullspace the orthogonal complement of the column space or the the row space? Thanks, sorry I don't have my textbook nearby.