91Ó°ÊÓ

Find the sum of the vectors and illustrate the sum geometrically. $$\mathbf{u}=(4,-2), \mathbf{v}=(-2,-3)$$

Given the coordinate matrix of \(x\) relative to a (nonstandard) basis \(B\) for \(R^{n}\), find the coordinate matrix of \(x\) relative to the standard basis. $$\begin{aligned}&B=\\{(4,0,7,3),(0,5,-1,-1),(-3,4,2,1)\\\&(0,1,5,0)\\}\\\&[\mathbf{x}]_{B}=\left[\begin{array}{r}-2 \\\3 \\\4 \\\1\end{array}\right]\end{aligned}$$

Describing the Additive Inverse In Exercises \(7-12\), describe the additive inverse of a vector in the vector space. $$ M_{1,4} $$

Finding a Basis for a Row Space and Rank In Exercises \(5-12\), find (a) a basis for the row space and (b) the rank of the matrix. $$ \left[\begin{array}{rrrr} -2 & -4 & 4 & 5 \\ 3 & 6 & -6 & -4 \\ -2 & -4 & 4 & 9 \end{array}\right] $$

Describing the Additive Inverse In Exercises \(7-12\), describe the additive inverse of a vector in the vector space. $$ M_{1,4} $$

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

Determine whether the set \(S\) spans \(R^{2} .\) If the set does not span \(R^{2},\) then give a geometric description of the subspace that it does span. \(S=\\{(5,0),(5,-4)\\}\)

In Exercises \(5-12,\) find (a) a basis for the row space and(b) the rank of the matrix. \(\left[\begin{array}{rrrr}-2 & -4 & 4 & 5 \\ 3 & 6 & -6 & -4 \\ -2 & -4 & 4 & 9\end{array}\right]\)

Determine which functions are solutions of the linear differential equation. $$ y^{\prime \prime}-y^{\prime}-12 y=0 $$ (a) \(e^{-4 x}\) (b) \(e^{4 x}\) (c) \(e^{-3 x}\) (d) \(e^{3 x}\)

Find the coordinate matrix of \(\mathbf{x}\) in \(R^{n}\) relative to the basis \(B^{\prime}\) $$B^{\prime}=\\{(4,0),(0,3)\\}, \mathbf{x}=(12,6)$$