If 𝐍(A) is the zero vector, what vectors are in the nullspace of B=[A A A] ?

Name: Problem 10, Chapter 1: Linear Algebra And Learning from Data
Uploaded: 2024-03-28T15:59:39-08:00
Duration: 1 min 31 s
Channel: Kratika Bhadauria
Description: If 𝐍(A) is the zero vector, what vectors are in the nullspace of B=[A A A] ?

step 1 Okay, so what is the null space? The null space of a matrix A is the set of all vectors x such t

If 𝐍(A) is the zero vector, what vectors are in the...

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Key Concepts

Nullspace (Kernel) of a Matrix

The nullspace of a matrix is the set of all vectors that are mapped to the zero vector by the matrix. It is a fundamental concept in linear algebra used to determine properties such as injectivity. If the nullspace contains only the zero vector, the transformation is one-to-one, meaning no nonzero vector is sent to zero.

Block Matrix Construction and Its Effect

A block matrix, such as one formed by concatenating matrices horizontally (e.g., B = [A A A]), creates a new matrix acting on vectors that are partitioned in accordance with the blocks. The action of this block matrix on an input vector involves applying each block to its corresponding segment and combining the results, which can often lead to constraints that relate the different segments.

Sum Condition for Nullspace Membership

When the same matrix A appears multiple times in a block matrix, the product of the block matrix with a partitioned vector often reduces to the matrix A acting on the sum of its segments. With the nullspace of A being trivial (only the zero vector), for the product to be zero the sum of these segments must be zero. This sum condition characterizes the nullspace of the block matrix.

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