Prove that A 𝐯=0 for every vector 𝐯 (with the appropriate number of entries) if and only if A=O

step 1 For this problem, we're given a matrix A, which is an M by N matrix, and a matrix D, which is a

Prove that A 𝐯=0 for every vector 𝐯 (with the appropriate...

Prove that $A \mathbf{v}=\mathbf{0}$ for every vector $\mathbf{v}$ (with the appropriate number of entries) if and only if $A=\mathrm{O}$ is the zero matrix. Hint: If you are stuck, first try to find a proof when $A$ is a small matrix, e.g., of size $2 \times 2$.

Key Concepts

Linear Transformations

Every matrix represents a linear transformation. A key property of linear transformations is that if a transformation sends every vector to the zero vector, then the transformation itself must be the zero transformation. This conceptual framework underlies the equivalence between the annihilation of all vectors and the matrix being the zero matrix.

Standard Basis

The standard basis comprises vectors with one entry equal to one and the rest zero. Using these vectors in matrix-vector multiplication isolates individual columns of the matrix. If the product is the zero vector for each standard basis vector, then every column of the matrix is the zero column.

Biconditional Proof (If and Only If)

A biconditional proof involves showing two directions: first, if the matrix is the zero matrix then its action on any vector yields the zero vector; and second, if the matrix sends every vector to the zero vector, then each entry of the matrix must be zero. This dual approach is essential for establishing equivalence in mathematical proofs.

Zero Matrix

The zero matrix is defined as a matrix in which every entry is zero. This is a fundamental concept because, by definition, multiplying any vector by the zero matrix yields the zero vector. Establishing that a matrix is the zero matrix directly shows that its effect on any vector is null.

Matrix-Vector Multiplication

Matrix-vector multiplication is a process where each component of the resulting vector is the dot product of the corresponding row of the matrix with the vector. If every such dot product is zero regardless of the vector chosen, it implies that every entry in the matrix must be zero.

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