Find 3u - 2v.

u = (-1, 2, 1), v = (0, 1, 1)

Write each vector as a linear combination of the vectors in $S$ (if possible).

$$S=\{(1,2,-2),(2,-1,1)\}$$

$$\mathbf{u}=(1,-5,-5)$$

Determine which functions are solutions of the linear differential equation.

$$y^{\prime \prime \prime}+y=0$$

$$2e^{-x}$$

Write each vector as a linear combination of the vectors in $S$ (if possible).

$$S=\{(1,2,-2),(2,-1,1)\}$$

$$\mathbf{w}=(-1,-22,22)$$

Explain why S is not a basis for $M_{2,2}$ $$ S=\left\ \left[\begin{array}{ll} {1} & {0} \\ {0} & {1} \end{array}\right],\left[\begin{array}{ll} {0} & {1} \\ {1} & {0} \end{array}\right],\left[\begin{array}{ll} {1} & {1} \\ {0} & {0} \end{array}\right]\right\} $$

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

Determine which functions are solutions of the linear differential equation.

$$y^{\prime \prime \prime}+y=0$$

$$e^{-2x}$$

Find u - v.

u = (-1, 2, 1), v = (0, 1, 1)

Find the nullspace of the matrix.

$$A=\left[\begin{array}{lll} {1} & {4} & {2} \\ {0} & {0} & {1} \end{array}\right]$$

Determine whether the set, together with the standard 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 $2 \times 2$ nonsingular matrices

Determine which functions are solutions of the linear differential equation.

$$y^{\prime \prime \prime}+y=0$$

$$e^{-x}$$

Write each vector as a linear combination of the vectors in $S$ (if possible).

$$S=\{(1,2,-2),(2,-1,1)\}$$

$$\mathbf{v}=(-2,-6,6)$$

Determine which functions are solutions of the linear differential equation.

$$y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0$$

$a)$ $x$ $\hspace{5mm}$ $b)$ $e^x$ $\hspace{5mm}$ $c)$ $e^{-x}$ $\hspace{5mm}$ $d)$ $x e^{-x}$

Find 2v.

u = (-1, 2, 1), v = (0, 1, 1)