# Proving two Spans of Vectors are Equal Linear Algebra Proof | Summary and Q&A

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December 16, 2015
by
The Math Sorcerer
Proving two Spans of Vectors are Equal Linear Algebra Proof

## TL;DR

This video provides a step-by-step proof that two spans in linear algebra are equal.

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### Q: What is the purpose of this video?

The purpose of this video is to demonstrate a direct proof of the equality of two spans in linear algebra without relying on theorems.

### Q: How is the equality of the spans shown?

The method of double inclusion is used, where it is shown that each vector in one span can be written as a linear combination of the vectors in the other span, and vice versa.

### Q: How is a vector represented as a linear combination?

A vector X can be written as a linear combination of vectors by multiplying each vector by a scalar and adding them together.

### Q: What is the significance of proving the equality of spans?

Proving the equality of spans is important in understanding linear independence and the relationships between vectors in linear algebra.

## Summary & Key Takeaways

• The video aims to prove that two sets of linear combinations of vectors are equal.

• Using the method of double inclusion, the proof shows that both sets are subsets of each other.

• The first direction is straightforward, while the second direction involves carefully rewriting one vector as a linear combination of the others.