Show that (1, 0), (i, 0) ∈ C2 are linearly independent over R and linearly dependentover C

Prove that the set of vectors (1, 2), (3, 4) is linearly independent and spanning in R2

Let and be any three non-zero vectors. Then, is independent ofx

To determine if the vectors \( u \), \( v \), and \( w \) are linearly dependent, we need to check if there exists a non-trivial linear combination of these vectors that equals the zero vector. In other words, we need to find scalars \( a \), \( b \), and \( c \) (not all zero) such that: \[ a \cdot u + b \cdot v + c \cdot w = 0 \] Given vectors: \[ u = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \quad v = \begin{pmatrix} 2 \\ -1 \end{pmatrix}, \quad w = \begin{pmatrix} -5 \\ 1 \end{pmatrix} \] We have already found that \( w \) can be written as a linear combination of \( u \) and \( v \) with specific scalars \( a = -1 \) and \( b = -2 \): \[ w = -1 \cdot u - 2 \cdot v \] This equation shows that there is a non-trivial linear combination of \( u \), \( v \), and \( w \) that equals the zero vector: \[ -1 \cdot u - 2 \cdot v + 1 \cdot w = 0 \] Since we have found a non-trivial solution where the scalars are not all zero, the vectors \( u \), \( v \), and \( w \) are linearly dependent. The fact that \( w \) can be expressed as a linear combination of \( u \) and \( v \) is sufficient to establish their linear dependence.

If X and Y are subsets of a vector space, and either X or Y is linearly dependent, show thatX ∪ Y is linearly dependent.

3. (a) If X and Y are subsets of a vector space, and either X or Y is linearly dependent, show thatX ∪ Y is linearly dependent. [6 marks](b) Give an example of a vector space V with subsets X and Y , such that X and Y are linearlyindependent, but X ∪ Y is linearly dependent.