Curated OER
Linear Systems: Vector Spaces and Subspaces
In this vector space learning exercise, students find the column space of given matrices, the identify the vector space and determine if the matrix is invertible. This four-page learning exercise contains approximately six multi-step...
Curated OER
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.
Curated OER
Introduction to Vector Spaces
In this vector spaces learning exercise, students identify the vector in a given set. They explore addition and scalar multiplication in their vectors. This two-page learning exercise contains 13 problems.
Curated OER
Linear Systems: Vector Spaces and Subspaces
In this vector spaces and subspaces worksheet, students review column spaces of matrices. They determine if the set of all 2 by 2 matrices is a vector space. Students determine if the set of all invertible 2 by 2 matrices is a vector...
Curated OER
Linear Systems: Vector Spaces of a Matrix
In this vector space matrix activity, students decompose given vectors into two orthogonal pieces--one from the nullspace and another from the row space. this two-page activity contains examples and approximately five problems.
Curated OER
Linear Systems: Introduction to Vector Space
In this vector space worksheet, students find the column spaces of given matrices. This two-page worksheet contains examples, theorems, explanations, and approximately three problems.
Curated OER
Linear Systems: Vector Spaces
For this vector space worksheet, students explore linearly independent vectors. They identify the row space, null space and subspaces of a given matrix. This four-page worksheet contains approximately eight problems.
Curated OER
Linear Systems: Vector Spaces
For this vector spaces worksheet, students explore the column and rowspace of a matrix. They find the basis of subspaces associated with a given matrix. Students observe the relationship between vectors and the nullspace. This...
EngageNY
Coordinates of Points in Space
Combine vectors and matrices to describe transformations in space. Class members create visual representations of the addition of ordered pairs to discover the resulting parallelogram. They also examine the graphical representation of...
EngageNY
Solving Equations Involving Linear Transformations of the Coordinate Plane
How can matrices help us solve linear systems? Learners explore this question as they apply their understanding of transformation matrices to linear systems. They discover the inverse matrix and use it to solve the matrix equation...
Curated OER
Introduction to Linear Algebra 2.1
In this linear equation worksheet, students identify the vector space of a given matrices and find the appropriate system of equations for each of four solutions. Then they solve the system and write their conclusion.
Curated OER
Linear Systems: Independence, Basis and Dimension
In this basis and dimension worksheet, students explore vector spaces associated with rectangular matrices. The two-page worksheet contains approximately 20 short answer questions.
Curated OER
Linear Systems: The Column Space
In this column space worksheet, students find the column spaces of given matrices. This two-page worksheet contains examples of and explanations. One problem is provided for students to complete.
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Linear Systems: General Solutions
In this linear system worksheet, students explore the characteristics of given matrices. They multiply matrices and determine pivots and multiply solutions. This four-page worksheet contains approximately seven problems.
EngageNY
Matrix Multiplication Is Not Commutative
Should matrices be allowed to commute when they are being multiplied? Learners analyze this question to determine if the commutative property applies to matrices. They connect their exploration to transformations, vectors, and complex...
Curated OER
Linear Systems: Fundamental Theorem of Linear Algebra
In this linear algebra instructional activity, students complete matrix multiplication. They find the row space and nullspace of given matrices. This two-page instructional activity contains examples, explanations, and approximately...
Curated OER
Linear Systems: Linear Equations, Vectors and Matrices
In this matrices worksheet, students write matrix notation to represent linear systems. This four-page worksheet contains approximately 15 problems.
Curated OER
Linear Systems: Linear Equations
In this linear systems worksheet, students use a matrix to represent linear systems. This four page worksheet provides examples, explanations and approximately eight problems to complete.
Curated OER
Linear Systems: Review of Vectors
For this review of vectors worksheet, students 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.
Curated OER
Complex Analysis: Linear Transformations
In this linear transformation worksheet, students find a series of linear transformations and sketch them. This four-page worksheet contains approximately six problems.
Curated OER
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.
Curated OER
Linear Systems: Fundamental Theorem
In this fundamental theorem worksheet, student explore the nullspace and column space of given matrices. They explore the vectors needed to make it linearly independent and construct free variable columns. This two-page worksheet...
Curated OER
Linear Systems: Column Space and Nullspace of a Matrix
In this matrix worksheet, students find the column space and nullspace of a matrix. They give examples of vectors. This two-page worksheet contains four problems.
EngageNY
First-Person Computer Games
How do graphic designers project three-dimensional images onto two-dimensional spaces? Scholars connect their learning of matrix transformations to graphic design. They understand how to apply matrix transformations to make...