Is it true these are theorems in Linear Algebra?

Click For Summary
SUMMARY

The discussion centers on theorems in Linear Algebra, specifically regarding transition matrices and the relationships between row and column spaces of matrices. It is established that if each vector in basis B1 is a scalar multiple of a vector in basis B2, then the transition matrix PB1→B2 is diagonal. Additionally, it is confirmed that for n × n invertible matrices A and B, the row space of the product AB coincides with the row space of B. Furthermore, if two n × n matrices A and B share the same row space, they also have the same column space.

PREREQUISITES
  • Understanding of vector spaces and bases in Linear Algebra
  • Familiarity with transition matrices and their properties
  • Knowledge of row space and column space concepts
  • Experience with matrix multiplication and properties of invertible matrices
NEXT STEPS
  • Study the properties of transition matrices in Linear Algebra
  • Learn about the implications of row space and column space equivalence
  • Explore the characteristics of skew-symmetric matrices
  • Investigate proofs related to vector space relationships and matrix operations
USEFUL FOR

Students and professionals in mathematics, particularly those studying Linear Algebra, as well as educators seeking to clarify foundational concepts related to matrices and vector spaces.

Logan Land
Messages
83
Reaction score
0
If each vector in basis B1 is scalar multiple of some vector in basis B2 then transition matrix PB1→B2 is diagonal.The column space of matrix A is the set of solutions of Ax = b.If A is n × n invertible matrix and AB is defined then row space of AB coincides with row space of B.Column space of skew symmetric matrix coincides with its row space.If A and B are n×n matrices that have the same row space, then A and B have the same column space.
 
Physics news on Phys.org
LLand314 said:
If each vector in basis B1 is scalar multiple of some vector in basis B2 then transition matrix PB1→B2 is diagonal.The column space of matrix A is the set of solutions of Ax = b.If A is n × n invertible matrix and AB is defined then row space of AB coincides with row space of B.Column space of skew symmetric matrix coincides with its row space.If A and B are n×n matrices that have the same row space, then A and B have the same column space.
Hello Lland314,

Can you please post some details on your attempt on these and where exactly you are stuck?

Also, try not posting more than 2 questions in a single thread.
 
LLand314 said:
If each vector in basis B1 is scalar multiple of some vector in basis B2 then transition matrix PB1→B2 is diagonal.If A and B are n×n matrices that have the same row space, then A and B have the same column space.

Actually I am having issue attempting to prove the first quoted question as true, as such i believe it is false. But intuitively I think it is true.

As for the second quoted question the answer I am unsure if he meant an mxn or actually an nxn matrix
 

Similar threads

Undergrad Row space, Column space, Null space & Left null space
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Dimension and solution to matrix-vector product
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Index notation of vector rotation
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Confused about small detail in rank-nullity theorem
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad If L is diagonalizable, algebraic & geometric multiplicities are equal
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Passive Transformation and Rotation Matrix
  • · Replies 1 ·
Replies
1
Views
4K
High School Is it invalid to redefine the sgn function in this way in a proof?
  • · Replies 15 ·
Replies
15
Views
5K