Question

Consider the stacked vectors ai ak bi where a... ak are n-vectors and b counter example. bz are m-vectors. For each case, either prove or provide a (a) Suppose ai..... ak are linearly independent (we make no assumptions about bi..... bi). Can we conclude that the stacked vectors c . c are linearly indepndent? (b) Suppose a.... ak are linearly dependent (we make no assumptions about bi..... bi). Can we conclude that the stacked vectors c..ck are linearly depndent?

Answer 1

Answer:

(a) Yes,we can conclude that the vectors C₁,...,C₀ are linearly independent.

(b) No,the vectors C₁,...,C₀ can also be linearly independent. For example, take.

Step-by-step explanation:

Note that I'm using "o" instead of "k" for easier computation ∴ C₀ = Ck

Proof by letting ∝₁,...,∝₀ be scalars such that ∝₁c₁+...+∝₀c₀ = 0 are linearly independent.  see equation in picture.

See attached