Are the Columns of Matrix A Linearly Independent?

  • Context: Undergrad 
  • Thread starter Thread starter scholesmu
  • Start date Start date
  • Tags Tags
    Vector
Click For Summary
SUMMARY

The discussion focuses on the linear independence of the columns of matrix A, defined as A = [[1, 3, 0], [0, -4, 4], [3, 4, 5]]. The columns of A are determined to be linearly independent through the application of the definition of linear independence, which involves checking if the only solution to the equation Ax = 0 is the trivial solution. Additionally, the discussion touches on the properties of vector spaces, specifically addressing the zero vector multiplication and the subspace characteristics of diagonal matrices within the set of 2x2 matrices.

PREREQUISITES
  • Understanding of linear independence in vector spaces
  • Familiarity with matrix operations and properties
  • Knowledge of vector space definitions and subspaces
  • Basic proficiency in linear algebra concepts
NEXT STEPS
  • Study the concept of linear transformations and their relation to matrix columns
  • Learn about the rank and nullity of matrices in linear algebra
  • Explore the properties of subspaces in vector spaces, particularly in relation to diagonal matrices
  • Investigate the implications of the zero vector in vector space operations
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, matrix theory, and vector spaces. This discussion is beneficial for anyone seeking to deepen their understanding of linear independence and matrix properties.

scholesmu
Messages
3
Reaction score
0
Q1:
Let A =
1 3 0
0 -4 4
3 4 5

a) Find Cartesian equations for Col(A) and Null(A).
b) Are the columns of A linearly independent? Give full reasons for your answer (based on
the definition of linear independence).

Q2: Prove that in any vector space k × 0 = 0, where k is any real number and 0 is the zero vector.

Q3:
The set M2,2 of 2 × 2 matrices, with real entries, is a vector space.
The set of diagonal matrices D =
[(a 0
0 b) | a, b 2 E R is a subset of M2,2
a) Write down two particular matrices which belong to D, and two particular matrices which
belong to M2,2 but not to D.
b) Prove that D is a subspace of M2,2
 
Physics news on Phys.org

Similar threads

Undergrad Row space, Column space, Null space & Left null space
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Confused about small detail in rank-nullity theorem
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Why Use Gram-Schmidt to Make a Unitary Matrix?
  • · Replies 14 ·
Replies
14
Views
4K
Undergrad Characterizing linear independence in terms of span
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Dimension and solution to matrix-vector product
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Linear independence of three vectors
  • · Replies 45 ·
2
Replies
45
Views
5K
Undergrad Proof by induction of block diagonal decomposition of a matrix
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Vector subspace and basis vectors in the context of data science
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad 2 and 3 dimensional invariant subspaces of R4
  • · Replies 1 ·
Replies
1
Views
3K