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Linear Systems: Nullspace Matrix and Rank
In this nullspace matrix worksheet, students observe the rank and number of independent rows and columns a matrix has. They compute the product of matrices and describe the column space of each. This two-page worksheet contains...
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Linear Systems: Least Squares Approximations
In this least squares approximation worksheet, students use linear algebra to find the least square error and the equation of a line for given points. They find the equation of a parabola through provided points. This four-page...
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Linear Systems: Orthogonal Matrices
In this orthogonal matrices learning exercise, students explore characteristics of an orthogonal matrix. They use Gram-Schmidt to convert linearly independent vectors to orthonormal vectors. This three-page learning exercise contains...
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Linear Systems: Projections
In this projection worksheet, students determine the projections of a given matrix. This four-page worksheet contains explanations and examples and approximately ten problems to complete.
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Linear Systems: Review of Vectors
In this review of vectors worksheet, learners explore vectors and their uses. They multiply and subtract vectors and identify the linear combinations of given matrices. This two-page worksheet contains approximately ten problems.
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Linear Systems: Graphical Interpretation of Vectors
In this graphical interpretation of vectors activity, students execute operations of addition and multiplication by a scalar. They sketch results and determine the commutativity of vector addition. This three-page activity contains...
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Linear Systems: Least Squares Approximations
In this least squares approximation instructional activity, students determine the number of solutions a given linear system has. They find the least square error and the equation of a line of best fit for a set of three points. Students...
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Linear Systems: Graphical Interpretation of Vectors and Linear Systems
In this interpretation of vectors and linear systems worksheet, students describe the geometric appearance of two vectors. They add the two vectors and sketch the sum as well as the vectors to determine the commutativity of vector...
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Linear Systems: Matrix Operations
For this matrix operations worksheet, students perform matrix addition and multiplication. This two-page worksheet contains six problems. Definitions, examples and explanations are also provided.
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Linear Systems: Using Matrix Operations
In this linear systems learning exercise, students use matrix multiplication as an elimination technique to solve linear systems. They compare products and explore trends. This four-page learning exercise contains explanations,...
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Linear Systems: LU Decomposition
In this LU decomposition worksheet, students multiply matrices, identify the elimination matrices, and the inverse of a matrix. This three-page worksheet contains approximately six problems.
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Linear Systems: Matrix Inverse
In this linear systems worksheet students multiply matrices and find the coefficient matrix in the upper-triangular form. This four-page worksheet contains approximately four problems, with examples and explanations.
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Linear Systems: The Determinant
In this determinant activity, students find the determinants of matrices. They identify patterns and their relationship to the canonical matrix. This two-page activity contains approximately nine problems, as well as examples and...
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Linear Systems: Permutations and Transposes
In this permutations worksheet, students identify a transpose matrix, and create a permutation matrix. This three-page worksheet contains six problems.
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Linear Systems: Eigenvalues and Eigenvectors
In this "Eigenvalues and Eigenvectors" worksheet, students find the eigenvalue of a matrix and explore the properties of eigenvalues with a matrix. This two page worksheet contains two multi-step problems.
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Linear Systems: Other Methods of Computing Determinants
In this computing determinants worksheet, students explore methods of computing determinants. They complete determinant problems by using the co-factor method, the "big formula" and the use of product of the pivots. This two-page...
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Linear Systems: Applications of Determinants
In this determinant activity, students explore determinants, area and volume. This two-page activity contains explanations and examples. It provides two problems for students to solve.
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Linear Systems: Geometric Interpretation of Eigenvalues
In this geometric interpretation worksheet, students prove the exponentiation of a matrix and the Eigenvalues of an inverse. This two-page worksheet contains explanations, examples, theorems and two problems for students to solve.
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Linear Systems: Applications of Eiginvalues
In this Eigenvalue and Eigenvector worksheet, students examine the applications of matrix exponentiation. This four-page worksheet provides explanations and examples, as well as three problems to complete.
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Linear Systems: Vector Spaces and Subspaces
In this vector space worksheet, students find the column space of given matrices, the identify the vector space and determine if the matrix is invertible. This four-page worksheet contains approximately six multi-step problems.
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Linear Systems: Reduced Row Echelon Matrix
In this echelon matrix activity, students find the nullspace of matrices by finding the reduced row echelon. This two-page activity contains explanations, examples and six problems to complete.
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Linear Systems: Orthogonality of Vector Spaces of a Matrix
In this vector spaces of a matrix worksheet, students decompose any vector into two orthogonal pieces. They identify the nullspace, row space, and find a reduced echelon. This two-page worksheet contains examples and two problems.
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Linear Systems
For this multiplication of vectors worksheet, students prove that the dot product operation is commutative. They normalize a vector so it has the same direction with a magnitude of one. Students use the angle formula to confirm...
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Linear Systems: Identifying Pivots and Multipliers
In this identifying pivots worksheet, students use algebra to find equivalent systems of equations. This four-page worksheet contains examples, explanations and approximately ten problems.