Unit 4: Vector Spaces and the Four Fundamental Subspaces
Every matrix carries four hidden spaces around with it — the column space, null space, row space, and left null space.
15 topics · 58 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
What a Vector Space Is
- How it worksHow do the rules for adding and scaling decide whether a collection of objects forms a vector space?
- Common mistakeWhy isn't a vector space required to be made of arrows or lists of numbers?
- Going furtherWhat happens to the vector space of all polynomials once you only allow degree 2 or less?
Your turn1 of 2
Matrices, polynomials, and geometric arrows all count as vector spaces. What makes them all the exact same KIND of mathematical object?Subspaces of a Vector Space
- How it worksHow do closure under addition and closure under scaling decide whether a subset is actually a subspace?
- Worked exampleShow that the set of vectors (x, y) with y equal to 2x forms a subspace of the plane.
- Going furtherWhat happens when you take the union of two different lines through the origin — does it stay a subspace?
- Common mistakeWhy isn't a line that misses the origin ever a subspace?
Your turn1 of 2
The set of points (x, y) with y = 2x + 4, where 4 is nonzero, does not pass through the origin. Why can this never be a subspace of the plane, regardless of what k and c are?The Column Space of a Matrix
- How it worksHow does the column space collect every vector you can reach by combining the columns of A?
- Worked exampleFind the column space of the matrix [[1, 2], [2, 4]].
- Common mistakeWhy isn't the column space of a matrix the same thing as the plain set of its columns?
- Going furtherWhat happens to the column space of a matrix when you add an extra column that's a combination of the others?
Your turn1 of 2
Why isn't the column space of a matrix the same thing as the plain set of its columns?The Null Space of a Matrix
- How it worksHow do free variables in the reduced matrix generate every single vector in the null space?
- Worked exampleFind the null space of the matrix [[1, 2, 3], [2, 4, 6]].
- Worked exampleFind every vector that the matrix [[1, 1, 2], [2, 2, 4], [1, 0, 1]] sends to zero.
- Common mistakeWhy isn't the null space just the set of zero entries sitting inside a matrix?
- Going furtherWhat happens to the null space of a matrix once it has full column rank?
Your turn1 of 2
A matrix has first row (1, 3, 2) and its second row is exactly double the first row (so its rank is 1, with 2 free variables among x, y, z in its null space). Find the value of x in the null-space vector where y = 1 and z = 0.Solving Ax = 0
- How it worksHow does setting each free variable to 1 in turn generate the special solutions that span the null space?
- Worked exampleFind the special solutions spanning the null space of the matrix [[1, 2, 0, 3], [0, 0, 1, 4]].
- Ties togetherHow does the null space of a matrix connect directly to whether its columns are linearly independent?
Your turn1 of 2
A matrix reduces to [[1, 3, 0, 2], [0, 0, 1, 7]]. Its null space is spanned by two special solutions, one for each free variable (x2 and x4). Find the x1 value of the special solution with x2 = 0 and x4 = 1.The Complete Solution to Ax = b
- Common mistakeWhy doesn't a system with a nonzero null space mean there is no solution to Ax = b?
- How it worksHow does adding any vector from the null space to one particular solution generate every other solution?
- Worked exampleFind the complete solution to the system x + 2y = 3, 2x + 4y = 6.
- Going furtherWhat happens to the complete solution once the null space of A contains only the zero vector?
Your turn1 of 2
The system x + 2y = 3, with its second equation being a multiple of the first (so infinitely many solutions lie along one line), has null-space direction (-2, 1) and particular solution (3, 0). Find the x-coordinate of the general solution at parameter t = 5.Linear Independence
- How it worksHow does forcing c1v1 + c2v2 + ... + cnvn = 0 to have only the all-zero solution test for independence?
- Worked exampleShow that the vectors (1, 0, 1), (0, 1, 1), and (1, 1, 0) are linearly independent.
- Common mistakeWhy doesn't checking that no two vectors are multiples of each other prove three or more vectors are independent?
- Going furtherWhat happens to independence the moment a fourth vector joins three vectors that already span all of 3-space?
Your turn1 of 2
The vectors (1, 0, 2), (0, 1, 2), and (1, 5, 0) are independent exactly when the determinant of the matrix they form is nonzero. Find that determinant.A Basis of a Vector Space
- How it worksHow does a basis have to be both independent and span the whole space at the same time?
- Worked exampleShow that (1, 0) and (1, 1) form a basis for the plane.
- Common mistakeWhy doesn't a spanning set automatically count as a basis?
- Going furtherWhat happens when you try to build a basis for the plane out of three vectors instead of two?
Your turn1 of 2
Find ad - bc for the vectors (4, 3) and (8, -4), which form a basis of the plane exactly when this value is nonzero.The Dimension of a Vector Space
- How it worksHow does counting the vectors in any one basis give you the dimension of the whole space?
- Worked exampleFind the dimension of the subspace of the plane spanned by (2, 4) and (1, 2).
- Common mistakeWhy isn't the dimension of a vector space just how many numbers it takes to write one vector down?
- Ties togetherHow does the dimension of the column space connect back to the rank of a matrix?
Your turn1 of 2
Find the dimension of the column space of a matrix that has 6 pivot columns after row reduction.Row Rank Equals Column Rank
- How it worksHow does elimination preserve the row space while collapsing dependent columns, proving the two ranks must agree?
- Worked exampleShow that row rank equals column rank for the matrix [[1, 2, 3], [2, 4, 7]].
- Common mistakeWhy isn't the row space of a matrix literally equal to its column space, even though their dimensions always match?
- Going furtherWhat happens to this row-rank-equals-column-rank equality when the matrix is rectangular instead of square?
Your turn1 of 2
Show that row rank equals column rank for the matrix [[1, 3, 2], [0, 1, 7]]. A general vector in the row space is s(1, a, b) + t(0, 1, c) for scalars s and t. Find its second entry when s = -3 and t = -5.The Row Space and Left Null Space
- How it worksHow does the row space collect every combination of the rows of A, the same way the column space combines columns?
- Worked exampleFind the row space and left null space of the matrix [[1, 2], [2, 4]].
- Common mistakeWhy isn't the row space of a matrix the same as its column space, even when they share the same dimension?
- Going furtherWhat happens to the left null space of a matrix that has full row rank?
Your turn1 of 2
A 2 by 2 matrix has first row (3, 2) and its second row is exactly 6 times the first row. Find the second entry of a left-null-space vector whose first entry is 1.The Rank of a Matrix
- How it worksHow does the number of pivots left after elimination tell you the rank of a matrix?
- Worked exampleFind the rank of the matrix [[1, 2, 3], [2, 4, 6], [1, 1, 1]].
- Common mistakeWhy isn't the rank of a matrix just the number of rows it has?
- Going furtherWhat happens to the rank of a matrix when you add a row that's a combination of the existing rows?
Your turn1 of 2
A matrix currently has rank 5. A new row is added that is NOT a linear combination of the existing rows. Find the new rank.The Rank-Nullity Theorem
- How it worksHow does the number of pivot columns plus the number of free columns always add up to the total column count?
- Worked exampleShow the rank-nullity theorem holds for the matrix [[1, 2, 3, 4], [2, 4, 6, 9]].
- Common mistakeWhy doesn't a bigger null space mean a matrix has to have a smaller column space?
Your turn1 of 2
A matrix has 7 columns and rank 2. Find its nullity — the dimension of its null space.Existence and Uniqueness via Rank
- How it worksHow does comparing rank(A) to the rank of the augmented matrix [A given b] tell you whether a solution exists at all?
- Worked exampleFind the number of solutions of the system x + y = 2, 2x + 2y = 5 by comparing the rank of A to the rank of the augmented matrix.
- Common mistakeWhy doesn't a system with more equations than unknowns automatically mean there is no solution?
Your turn1 of 2
A system has 7 unknowns. Its coefficient matrix A has rank 2, and the augmented matrix [A | b] also has rank 2 (so the system is consistent). Find the number of free variables in the solution.The Four Fundamental Subspaces
- How it worksHow do the column space, null space, row space, and left null space each live in either the input space or the output space?
- Worked exampleFind the dimensions of all four fundamental subspaces of the matrix [[1, 2, 3], [2, 4, 6]].
- Common mistakeWhy doesn't the row space of A end up equal to the column space of A, even though the two have the same dimension?
- Going furtherWhat happens to the four fundamental subspaces when the matrix A is square and invertible?
- Ties togetherHow does orthogonality between the four subspaces set up the picture used later for projections?
Your turn1 of 2
A matrix is 7 by 7 with rank 3. Find the dimension of its left null space.