91Ó°ÊÓ

Find the sum of the vectors and illustrate the indicated vector operations geometrically. $$\mathbf{u}=(1,3), \mathbf{v}=(2,-2)$$

Determine which functions are solutions of the linear differential equation. \(y^{\prime}-2 x y=0\) (a) \(y=3 e^{x^{2}}\) (b) \(y=x e^{x^{2}}\) (c) \(y=x^{2} e^{x}\) (d) \(y=x e^{-x}\)

Find the sum of the vectors and illustrate the indicated vector operations geometrically. $$\mathbf{u}=(-1,4), \mathbf{v}=(4,-3)$$

Find \((\mathrm{a})\) the rank of the matrix, \((\mathrm{b})\) a basis for the row space, and (c) a basis for the column space. $$\left[\begin{array}{ccc}2 & -3 & 1 \\ 5 & 10 & 6 \\ 8 & -7 & 5\end{array}\right]$$

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. $$S=\\{(2,0),(0,1)\\}$$

Find the coordinate matrix of \(\mathbf{x}\) in \(R^{n}\) relative to the basis \(B\) $$B=\\{(-6,7),(4,-3)\\}, x=(-26,32)$$

Describe the additive inverse of a vector in the vector space. $$C(-\infty, \infty)$$

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

\(W\) is not a subspace of the vector space. Verify this by giving a specific example that violates the test for a vector subspace (Theorem 4.5). \(W\) is the set of all vectors in \(R^{2}\) whose second component is 1

Find \((\mathrm{a})\) the rank of the matrix, \((\mathrm{b})\) a basis for the row space, and (c) a basis for the column space. $$\left[\begin{array}{rrrr}-2 & -4 & 4 & 5 \\ 3 & 6 & -6 & -4 \\ -2 & -4 & 4 & 9\end{array}\right]$$