91Ó°ÊÓ

Find the Wronskian for the set of functions. $$\left\\{1, e^{x}, e^{2 x}\right\\}$$

Find the vector \(\mathbf{v}\) and illustrate the indicated vector operations geometrically, where \(\mathbf{u}=(-2,3)\) and \(\mathbf{w}=(-3,-2)\) $$\mathbf{v}=\frac{1}{2}(3 \mathbf{u}+\mathbf{w})$$

Find the transition matrix from \(B\) to \(B^{\prime}\) by hand $$B=\\{(2,4),(-1,3)\\}, B^{\prime}=\\{(1,0),(0,1)\\}$$

Determine whether the set \(S\) spans \(R^{2}\). If the set does not span \(R^{2}\), give a geometric description of the subspace that it does span. $$S=\\{(-1,4),(4,-1),(1,1)\\}$$

Determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. The set of all third-degree polynomials with the standard operations

Find a basis for the subspace of \(R^{3}\) spanned by \(S\) $$S=\\{(4,4,8),(1,1,2),(1,1,1)\\}$$

Determine whether the set \(S\) spans \(R^{2}\). If the set does not span \(R^{2}\), give a geometric description of the subspace that it does span. $$S=\\{(-1,2),(2,-1),(1,1)\\}$$

Explain why \(S\) is not a basis for \(R^{3}\). $$S=\\{(2,1,-2),(-2,-1,2),(4,2,-4)\\}$$

Determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. The set of all fifth-degree polynomials with the standard operations

Find a basis for the subspace of \(R^{3}\) spanned by \(S\) $$S=\\{(1,2,2),(-1,0,0),(1,1,1)\\}$$