Linear Algebra Problems

In summary, the conversation discusses resources for linear algebra problems for students to practice and improve their understanding. Recommendations include math worksheets, Schaum's outline series, and Hefferon's free textbook with a solutions manual. The question of whether a specific operation in R^3 defines a vector space is also mentioned, along with the properties required for addition in a vector space. The Cambridge Linear Algebra course example sheets are suggested as a source for excellent questions.
  • #1
matqkks
285
5
Linear Algebra Problems
Are there any resources consisting of a collection of problems on linear algebra for students to practice? I am looking for good interesting problems which test students’ understanding. These questions or examples should be for teaching rather than just testing. The level of difficulty is first year undergraduate.
 
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  • #2
mathworksheetsgo.com
 
  • #3
Let me know if you find any matqkks. I too am interested.
 
  • #5
I am looking for help in this question."Rather than using the standard definitions of addition and scalar multiplication in R^3, suppose the operation is define as follows: (c1,y1,z1,) + (x2,y2,z2) = (x1+x2+1,y1+y2+1,z1+z2+1)

c(x,y,z) = (cx,cy,cz)
With this new definitions, is a vector space?
Justify your answers
 
  • #6
Determine whether or not this addition (the scalar multiplication defined is just the usual one on [itex]R^3[/itex]) has the properties required for addition in a vector space:
1) u+ v= v+ u (commutativity)
2) (u+ v)+ w= u+ (v+ w) (associativity)
3) There exist a specific vector, O, such that v+ O= O+ v= v for any vector v (O is NOT necessarily (0, 0, 0)).
4) For any vector v, there exist a vector, u, such that v+ u= u+ v= O.
5) For any number, a, and vectors u and v, a(u+ v)= au+ av. (Distributive)
 
  • #7
The examples sheets for the Cambridge Linear Algebra course contain some excellent questions. They can be found here.

In particular, the "Preliminary example sheet" (essentially a revision sheet) might be just what you seek.
 

What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of vectors, matrices, linear transformations, and systems of linear equations. It is used to solve problems related to geometry, physics, engineering, and many other fields.

What are some real-world applications of linear algebra?

Linear algebra has many practical applications, including image and signal processing, computer graphics, data analysis, and machine learning. It is also used in the fields of economics, statistics, and optimization.

What are the basic operations in linear algebra?

The basic operations in linear algebra include vector addition and multiplication, matrix addition and multiplication, and scalar multiplication. These operations are used to perform calculations and solve systems of linear equations.

How do I solve a system of linear equations?

To solve a system of linear equations, you can use various methods such as Gaussian elimination, Cramer's rule, or matrix inversion. These methods involve performing operations on the equations to isolate the variables and find their values.

What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important concepts in linear algebra. Eigenvalues represent the scaling factor of an eigenvector when it is transformed by a linear transformation. They are used in many applications, such as principal component analysis and image compression.

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