91Ó°ÊÓ

You are provided with the coordinate matrix of \(\mathbf{x}\) relative to a (nonstandard) basis \(B\). Find the coordinate vector of \(\mathbf{x}\) relative to the standard basis in \(R^{\prime \prime}\) $$B=\\{(0,0,0,1),(0,0,1,1),(0,1,1,1),(1,1,1,1)\\}$$ $$[\mathbf{x}]_{n}=[1,-2,3,-1]^{T}$$

Describe the zero vector (the additive identity) of the vector space. $$P_{3}$$

Use a directed line segment to represent the vector $$\mathbf{u}=(-3,-4)$$

Determine which functions are solutions of the linear differential equation. \(x^{2} y^{\prime \prime}-2 y=0\) (a) \(y=\frac{1}{x^{2}}\) (b) \(y=x^{2}\) (c) \(y=e^{x^{2}}\) (d) \(y=e^{-x^{2}}\)

Find \((\mathrm{a})\) the rank of the matrix, \((\mathrm{b})\) a basis for the row space, and (c) a basis for the column space. $$\left[\begin{array}{rrr}1 & -3 & 2 \\ 4 & 2 & 1\end{array}\right]$$

Verify that \(W\) is a subspace of \(V .\) In each case assume that \(V\) has the standard operations. \(W\) is the set of all functions that are continuous on \([0,1] . V\) is the set of all functions that are integrable on [0,1]

Verify that \(W\) is a subspace of \(V .\) In each case assume that \(V\) has the standard operations. \(W\) is the set of all functions that are differentiable on \([0,1] . V\) is the set of all functions that are continuous on [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. $$S=\\{(1,-1),(2,1)\\}$$

Find \((\mathrm{a})\) the rank of the matrix, \((\mathrm{b})\) a basis for the row space, and (c) a basis for the column space. $$\left[\begin{array}{rrr}1 & 2 & 4 \\ -1 & 2 & 1\end{array}\right]$$

Determine which functions are solutions of the linear differential equation. \(x y^{\prime \prime}+2 y^{\prime}=0\) (a) \(y=x\) (b) \(y=\frac{1}{x}\) (c) \(y=x e^{x}\) (d) \(y=x e^{-x}\)