Lecture 13: Equivalent Sets of Vectors

13.1 Basic Concepts

Concept One: Vector set equivalence and matrix equivalence are two different concepts;

• Matrix equivalence must be isomorphic, so the number of rows and columns must be equal;
• Vector sets are equivalent, but the number of vectors can be different;

Concept 2: If $A$ and $B$ have the same dimensions, then $A\cong B\Leftrightarrow r(A)=r(B)\Leftrightarrow PAQ=B$, where $P$ and $Q$ are invertible matrices of suitable sizes.

• Convert a matrix of equal rank into all matrices in its simplest form, which are equivalent;

Concept 3: For two sets of vectors in the same vector space:

• $\{\alpha_{1},\alpha_{2},\cdots,\alpha_{s}\}\cong\{\beta_{1},\beta_{2},\cdots,\beta_{t}\}$
• $\Leftrightarrow$ each vector set can be linearly represented by the other.
• $\Leftrightarrow$ the two sets have the same rank and one set can be linearly represented by the other.
• $\Leftrightarrow r\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{s}\right)=r\left(\beta_{1},\beta_{2},\cdots,\beta_{t}\right)=r\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta_{1},\beta_{2},\cdots,\beta_{t}\right)$.