# Proving Linear Independence of a List of Vectors.

The problem below has been taken from Klaus Janich's Linear Algebra textbook.

Let $$V$$ be a vector space over $$\mathbb{R}$$ and let $$a,b,c,d \in V$$. Suppose that:

$$v_1 = a + b + c + d$$

$$v_2 = 2a + 2b + c -d$$

$$v_3 = a + b + 3c - d$$

$$v_4 = a - c + d$$

$$v_5 = -b + c - d$$

Show that $$(v_1,v_2,\ldots,v_5)$$ is linearly dependent. There is an elegant way to do this.

I've come up with an argument and I just need someone to check if it works or not.

My Proof Attempt:

Suppose that the given list of vectors, which we denote by $$A$$ is linearly independent. Then, the list of vectors $$B = (a,b,c,d)$$ is also linearly independent. This follows from the fact that any linear combination of the vectors in $$A$$ can be written as a linear combination of the vectors in $$B$$.

So, $$A$$ is linearly independent and clearly spans $$L(A)$$. That is the linear hull of $$A$$. Hence, the vectors in $$A$$ form a basis for $$L(A)$$. However, it is also true that the vectors in $$B$$ form a basis for $$L(A)$$ as well. Therefore, $$L(A)$$ has two bases of different lengths.

That is a contradiction. Hence, the vectors in $$A$$ are not linearly independent.

Please give detailed feedback on the way I've written my proof up as well. If there's any way I can improve my literary style to match modern standards, I'd gladly take that way. Thank you in advance.

• Bump haha. Could anyone help? – Abhi Feb 23 at 3:52

You can prove a generalized version of this result. Precisely, given $$n$$ vectors $$\{v_{1},v_{2},\ldots,v_{n}\}$$ from the linear space $$V$$, any set of $$n+1$$ vectors $$\{w_{1},w_{2},\ldots,w_{n},w_{n+1}\}$$, where each $$w_{j}\in\text{Span}\{v_{1},v_{2},\ldots,v_{n}\}$$, is linear dependent.
Let $$\{v_{1},v_{2},\ldots,v_{n}\}\subset V$$. Since each $$w_{j}\in\text{Span}\{v_{1},v_{2},\ldots,v_{n}\}$$, the set $$\{w_{1},w_{2},\ldots,w_{n+1}\}$$ must be LD. Indeed, we have that $$\dim\text{Span}\{v_{1},v_{2},\ldots,v_{n}\} \leq n$$. If the vectors $$\{w_{1},w_{2},\ldots, w_{n+1}\}$$ were LI, then we would have that $$\dim\text{Span}\{w_{1},w_{2},\ldots,w_{n+1}\} = n+1$$, which is impossible, because $$\text{Span}\{w_{1},w_{2},\ldots,w_{n+1}\}\subseteq\text{Span}\{v_{1},v_{2},\ldots,v_{n}\}$$