If a 3 ×8 matrix A has rank 3, find dim Nul A, dim Row A and rank A^T .

Name: Problem 5, Chapter 4: Linear Algebra and Its Applications
Uploaded: 2020-09-19T14:30:19-08:00
Duration: 2 min 52 s
Channel: Subham Jyoti Mishra
Description: If a 3 ×8 matrix A has rank 3, find dim Nul A, dim Row A and rank A^T .

step 1 So in question we have 3 into 8 matrix and it's given that a b 3 into 8 matrix rank of a is give

If a 3 ×8 matrix A has rank 3, find dim Nul A, dim Row A and...

Key Concepts

Rank of a Matrix

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the row space or column space and is a measure of the matrix's nondegeneracy.

Nullspace (Kernel) and Nullity

The nullspace (or kernel) of a matrix is the set of all vectors that, when multiplied by the matrix, yield the zero vector. The nullity is the dimension of this nullspace, giving insight into the number of free parameters in the solutions of the corresponding homogeneous linear system.

Rank-Nullity Theorem

The Rank-Nullity Theorem establishes a fundamental relationship in linear algebra, stating that for any matrix, the sum of its rank and nullity equals the number of columns. This theorem is vital for understanding the solution space of linear systems and the structural properties of matrices.

Row Space

The row space of a matrix is the set of all linear combinations of its row vectors. Its dimension is equal to the rank of the matrix, and it plays a crucial role in analyzing the solvability and the solution structure of linear systems.

Transpose of a Matrix and Rank

The transpose of a matrix is formed by swapping its rows and columns. Importantly, the rank of a matrix is invariant under transposition, meaning that the rank of the matrix is equal to the rank of its transpose. This concept is essential for understanding properties of the matrix and its various subspaces.

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