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    • ▾Linear algebra
      • •Adding and subtracting matrices
      • •Adding vectors in polar form
      • •Applications of linear algebra
      • •Applications of vectors
      • •Area of a parallelogram using determinant
      • •Area of a triangle using determinant
      • •Are the vectors parallel, orthogonal, or neither?
      • ▸Basic transformation matrices
        • •Transforming a point
        • •Composition of transformations
      • ▸Basis
        • •Basis for a null space
        • •Basis for a span
      • ▸Cramer's rule
        • •Solving linear systems in two variables using Cramer's rule
        • •Solving linear systems in three variables using Cramer's rule
      • •Definition of \(\mathbb{R}^n\)
      • ▸Determinants
        • •Determinant using row echelon form
        • •Determinant using cofactors
        • ▸Determinant using row operations
          • •Determinant of a \(4{\times}4\) matrix using row operations
        • ▸Determinant of a \(3{\times}3\) matrix
          • •Determinant of a \(3{\times}3\) matrix using expansion by minors
          • •Determinant of a \(3{\times}3\) matrix using the diagonal method
        • •Determinant of a \(2{\times}2\) matrix
        • •Area of a triangle using determinants
        • •Intuition: Determinants
      • •Diagonalization
      • ▸Eigenvalues and eigenvectors
        • •Power method
        • •Shifted inverse power method
        • •Finding the eigenvalues and eigenvectors of \(2{\times}2\) matrices
        • •Finding the eigenvalues and eigenvectors of \(3{\times}3\) matrices
        • •Finding the eigenvalues and eigenvectors of \(4{\times}4\) matrices
      • •Elementary matrices
      • •Applications of the dot product
      • •Gaussian elimination
      • ▸Geometric proofs using vectors
        • •Proof: Law of cosines using the dot product
        • •Proof: Rhombus diagonals are perpendicular bisectors (linear algebra)
        • •Proof: Varignon's theorem (linear algebra)
        • •Proof: Thales's theorem using the dot product
      • •Gershgorin circle theorem
      • •History
      • •Length of a vector
      • •Linear combinations
      • ▸Linear transformations
        • •Linear transformations on \(\mathbb{R}^2\)
      • ▸Magnitude and direction form of vectors
        • •Lesson
        • •Practice
      • •Matrix inverses
      • ▸Matrix multiplication
        • •Intuition: Matrix multiplication
      • ▸Multiplying a matrix by a scalar
        • •Practice
      • •Multiplying a vector by a scalar
      • •Orthogonal vectors
      • •Parallel vectors
      • •Proof: Cauchy–Schwarz inequality
      • •Proof: Cross product distributes over addition
      • •Proof: Dot product formula
      • •Proof: Vector addition is commutative and associative
      • ▸Proof the cross product is distributive
        • •Algebraic proof the cross product is distributive
        • •Geometric proof the cross product is distributive
      • •Properties of matrix multiplication
      • •Properties of scalar multiplication
      • •Section formula
      • ▸Solving linear systems of equations
        • ▸Solving linear systems of equations using matrix inverses
          • •Solving linear systems of equations in two variables using matrix inverses
        • •Solving linear systems of equations in three variables using Gaussian elimination
        • •Consistency of a system of linear equations
      • •Gauss–Jordan elimination
      • •Solving linear systems of equations in two variables using Gauss–Jordan elimination
      • •Solving matrix equations
      • •Transpose and its properties
      • ▸Unit vectors
        • •Finding the unit vector
      • •Vector magnitude
      • ▸Vector projection
        • •Proof: Vector projection formula
      • ▸Vectors
        • ▸3D vectors
          • •3D vector operations
        • ▸Adding and subtracting vectors
          • •Adding vectors
          • •Subtracting vectors
        • ▸Components of vectors
          • •Lesson
          • •Practice
        • •Equivalent vectors
        • •Properties of vector addition
        • •Properties of vector addition and scalar multiplication
      • •What is an orthogonal set?
      • •What is a vector space?
      • •Zero matrix
      • •William Rowan Hamilton
      • ▸Solving linear systems of equations in three variables
        • •Solving linear systems of equations in three variables (word problems)
        • •Solving linear systems of equations in three variables by elimination
        • •Solving linear systems of equations in three variables by substitution
      • •Vector equations of lines
      • •Linear subspace
      • •Orthogonal complements
      • •Orthogonal projection
      • ▾The four fundamental spaces
        • •Column space
        • •Nullspace, left nullspace, and row space
      • •Van Aubel's theorem
      • •Visualizing a projection onto a plane
      • •Bretschneider's formula
     › Linear algebra › The four fundamental spaces

    Column space

    Students will learn that the column space of a matrix, is the span of all its vectors. For example, if \(A\) is an \(m \times n\) matrix, where

    $$A = [\vec{v_1}, \vec{v_1}, \ldots \vec{v_n}]$$

    Then it's column space is

    $$C(A) = \text{span}(\vec{v_1}, \vec{v_1}, \ldots \vec{v_n})$$

    Next, students will learn why the column space of a matrix is closed under scalar multiplication and vector addition. That is, if \(\vec{a}, \vec{b} \in C(A),\) then \(s\vec{a} \in C(A),\) and \(\vec{a} + \vec{b} \in C(A).\) Finally, students will learn that saying there is no solution to the system \(A\vec{x} = \vec{b},\) is equivalent to saying \(\vec{b} \not\in C(A).\) Here's a lesson on all that.