Let w1, w2, w3, u and v be vectors in Rn. Suppose the vectors u and vare in Span{w1, w2, w3}. Show that u + v is also in Span{w1, w2, w3}.

To show that the span of vectors \( u \), \( v \), and \( w \) is equal to \( \mathbb{R}^2 \), we need to demonstrate that any arbitrary vector in \( \mathbb{R}^2 \) can be expressed as a linear combination of \( u \), \( v \), and \( w \). Let's consider an arbitrary vector \( x \) in \( \mathbb{R}^2 \): \[ x = \begin{pmatrix} a \\ b \end{pmatrix} \] We need to find scalars \( \alpha \), \( \beta \), and \( \gamma \) such that: \[ x = \alpha u + \beta v + \gamma w \] Given vectors: \[ u = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \quad v = \begin{pmatrix} 2 \\ -1 \end{pmatrix}, \quad w = \begin{pmatrix} -5 \\ 1 \end{pmatrix} \] Substituting the given vectors into the equation, we get: \[ \begin{pmatrix} a \\ b \end{pmatrix} = \alpha \begin{pmatrix} 1 \\ 1 \end{pmatrix} + \beta \begin{pmatrix} 2 \\ -1 \end{pmatrix} + \gamma \begin{pmatrix} -5 \\ 1 \end{pmatrix} \] This leads to the following system of equations: \[ \alpha + 2\beta - 5\gamma = a \] \[ \alpha - \beta + \gamma = b \] Since we have two equations and three unknowns, we can express one of the unknowns, say \( \gamma \), in terms of \( \alpha \) and \( \beta \), and then solve for \( \alpha \) and \( \beta \) to express any vector \( x \) as a linear combination of \( u \), \( v \), and \( w \). However, we have already established that \( w \) can be written as a linear combination of \( u \) and \( v \) (from the previous answer, \( w = -1u - 2v \)). This means that any linear combination of \( u \), \( v \), and \( w \) can be reduced to a linear combination of just \( u \) and \( v \). Since \( u \) and \( v \) are linearly independent, they span \( \mathbb{R}^2 \), and thus the span of \( u \), \( v \), and \( w \) is also \( \mathbb{R}^2 \). In conclusion, the span of \( u \), \( v \), and \( w \) is \( \mathbb{R}^2 \) because \( u \) and \( v \) are sufficient to express any vector in \( \mathbb{R}^2 \), and \( w \) does not add any new dimension to the span.

Suppose {v1,v2,U3, v4} is a linearly dependent spanning set for a vector space V. Show that each w in V can be expressed in more than one way)as a linear combination of v1,...,v4.

Let W1 = L{(1, 1, 0), (−1, 1, 0)} and W2 = L{(1, 0, 2), (−1, 0, 4)}. Show that W1 +W2 = R3. Give an example of a vector v ∈ R3 such that v can be written in twodifferent ways in the form v = v1 + v2, where v1 ∈ W1, v2 ∈ W2

Let u, v and w be vectors in R3. Which of the following calculations is possible? (a) u + (v · w) (b) u · (v × w) (c) (u · v) × w (d) u × (v · w)

Let vector y= [ -1 3 5] and vector u= [-5 6 -7]. Write vector y as the sum of two orthogonal vectors, what are the vector x1 in span {u} and vector x2 orthogonal to vector u.