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Column Space Calculator (3×3 Matrix)

Find the column space of a 3×3 matrix. Enter 9 entries and get the RREF, pivot columns, basis vectors, and dimension (rank) instantly.

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Inputs

Row 1, Column 1 (a₁₁)

Row 1, Column 2 (a₁₂)

Row 1, Column 3 (a₁₃)

Row 2, Column 1 (a₂₁)

Row 2, Column 2 (a₂₂)

Row 2, Column 3 (a₂₃)

Row 3, Column 1 (a₃₁)

Row 3, Column 2 (a₃₂)

Row 3, Column 3 (a₃₃)

Dimension of Column Space (Rank)

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Get a plain-English breakdown of your result with practical next steps.

Dimension of Column Space (Rank)—

The formula

How the
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computed.

Understanding the Column Space of a 3×3 Matrix

The column space of a matrix A, written Col(A), is the set of all linear combinations of its column vectors. For a 3×3 matrix with real entries, this subspace lives within ℝ³ and captures the full range of output vectors the matrix can produce when multiplied by any vector in ℝ³.

The Core Formula

The dimension of the column space equals the rank of the matrix, which equals the number of pivot columns found after reducing the matrix to Reduced Row Echelon Form (RREF):

dim(Col(A)) = rank(A) = number of pivot columns in RREF(A)

For a 3×3 matrix, the rank — and therefore the dimension of the column space — takes a value of 0, 1, 2, or 3. A rank of 3 means the column space spans all of ℝ³; a rank of 0 occurs only for the zero matrix.

The Nine Matrix Entries

The 3×3 matrix A is defined by nine scalar entries arranged in three rows and three columns. Each column vector is formed by stacking the three entries in that column: column 1 is [a₁₁, a₂₁, a₃₁]ᵀ, column 2 is [a₁₂, a₂₂, a₃₂]ᵀ, and column 3 is [a₁₃, a₂₃, a₃₃]ᵀ. The column space is the span of these three vectors inside ℝ³.

How to Find the Column Space: Step-by-Step

Assemble the matrix: Arrange all nine entries into the 3×3 grid of A.
Row-reduce to RREF: Apply elementary row operations — row swap, scalar multiplication, and row addition — until the matrix reaches Reduced Row Echelon Form. The Georgia Tech Interactive Linear Algebra textbook on Basis and Dimension details how pivot positions emerge during this process.
Identify pivot columns: In the RREF, locate every column containing a leading 1 with all other entries in that column equal to zero.
Return to the original matrix: Go back to matrix A and extract the columns at the pivot positions found in the RREF. Row operations preserve the row space but alter the column vectors of the transformed matrix, so basis vectors must always come from the original A. The UC Davis Linear Algebra course notes reinforce this critical distinction.
State the basis and dimension: The selected original columns form a basis for Col(A). Their count equals rank(A) = dim(Col(A)).

Worked Example 1: Rank-2 Matrix

Let A = [[1, 2, 3], [2, 4, 6], [1, 1, 2]]. Row-reducing: subtract 2×R₁ from R₂ to obtain [0, 0, 0]; subtract R₁ from R₃ to obtain [0, −1, −1]; scale and swap to get [0, 1, 1]; then eliminate to reach RREF = [[1, 0, 1], [0, 1, 1], [0, 0, 0]]. Pivot columns are 1 and 2, so rank(A) = 2. The basis for Col(A) consists of the original columns c₁ = [1, 2, 1]ᵀ and c₂ = [2, 4, 1]ᵀ. The column space is a 2-dimensional plane through the origin in ℝ³.

Worked Example 2: Full-Rank Matrix

The 3×3 identity matrix I₃ = [[1,0,0],[0,1,0],[0,0,1]] is already in RREF. All three columns are pivot columns, so rank(I₃) = 3 and Col(I₃) = ℝ³. Every vector b in ℝ³ can be written as a linear combination of the standard basis vectors e₁, e₂, and e₃, confirming that Ax = b always has a unique solution when A = I₃.

Rank-Nullity Theorem Connection

The Rank-Nullity Theorem states that for any 3×3 matrix A: rank(A) + nullity(A) = 3. If rank(A) = 2, then dim(Null(A)) = 1 — exactly one independent direction in ℝ³ maps to the zero vector. This relationship ties the column space and null space together and determines the complete solution structure of Ax = b.

Practical Applications

Linear systems: The equation Ax = b has a solution if and only if b lies in Col(A). Checking whether the augmented matrix [A|b] introduces a new pivot immediately determines solvability.
Least-squares fitting: When b falls outside Col(A), the least-squares solution projects b onto Col(A) to minimize the residual error ||b − Ax||.
Data science (PCA): Principal Component Analysis identifies the most significant directions within the column space of a centered data matrix, enabling dimensionality reduction while preserving variance.
Control theory: The reachable state space of a linear dynamical system x(t+1) = Ax(t) + Bu(t) equals the column space of its controllability matrix.

Reference

Frequently asked questions

What is the column space of a 3×3 matrix?

The column space of a 3×3 matrix A, denoted Col(A), is the set of all vectors in ℝ³ expressible as a linear combination of A's three column vectors c₁, c₂, and c₃. Formally, Col(A) = {x₁c₁ + x₂c₂ + x₃c₃ : x₁, x₂, x₃ ∈ ℝ}. Depending on how many columns are linearly independent, the column space can be a single point (dimension 0), a line through the origin (dimension 1), a plane through the origin (dimension 2), or all of ℝ³ (dimension 3).

How does the column space calculator find the basis and dimension of a 3×3 matrix?

The calculator applies Gaussian elimination with back-substitution to reduce the entered 3×3 matrix to Reduced Row Echelon Form (RREF). It then identifies every pivot column — a column in the RREF containing a leading 1 with all other entries zero. The corresponding columns from the original (unreduced) matrix form the basis for Col(A), and the total count of pivot columns equals dim(Col(A)) = rank(A), a value between 0 and 3.

What is the difference between the column space and the null space of a 3×3 matrix?

The column space Col(A) contains all output vectors that A can produce — it is a subspace of the codomain ℝ³. The null space Null(A) contains all input vectors x for which Ax = 0 — it is a subspace of the domain ℝ³ representing directions that collapse to zero. The Rank-Nullity Theorem guarantees dim(Col(A)) + dim(Null(A)) = 3 for any 3×3 matrix, so a rank-2 matrix has a 1-dimensional null space paired with a 2-dimensional column space.

Can the column space of a 3×3 matrix span all of ℝ³?

Yes. When a 3×3 matrix A has rank 3 — meaning all three columns are linearly independent — then Col(A) = ℝ³. The RREF of such a matrix equals the 3×3 identity, with all three columns identified as pivot columns. A necessary and sufficient condition for this is a non-zero determinant, det(A) ≠ 0, providing a fast check for full rank. In this case, the equation Ax = b has a unique solution for every vector b in ℝ³.

What does the rank reveal about the column space of a 3×3 matrix?

The rank of a matrix directly equals the dimension of its column space. A 3×3 matrix with rank 1 has all columns proportional to each other, so Col(A) is a line through the origin in ℝ³. Rank 2 means exactly two columns are linearly independent, making Col(A) a plane through the origin. Rank 3 means all three columns are independent and Col(A) = ℝ³. Rank 0 occurs only for the zero matrix, whose column space contains only the zero vector.

How is the column space used to solve systems of linear equations?

The linear system Ax = b is consistent — having at least one solution — if and only if b belongs to Col(A). To verify this, form the augmented matrix [A|b] and row-reduce; if no new pivot appears in the last (augmented) column, then b is in Col(A) and the system is solvable. When b lies outside Col(A), the least-squares solution x* = (AᵀA)⁻¹Aᵀb produces the vector Ax* in Col(A) that minimizes the Euclidean distance ||b − Ax*||, giving the best approximate solution.