91Ó°ÊÓ

Find the transition matrix representing the change of coordinates on \(P_{3}\) from the ordered basis \(\left[1, x, x^{2}\right]\) to the ordered basis \\[ \left[1,1+x, 1+x+x^{2}\right] \\]

Let \(V\) be the set of all ordered pairs of real numbers with addition defined by \\[ \left(x_{1}, x_{2}\right)+\left(y_{1}, y_{2}\right)=\left(x_{1}+y_{1}, x_{2}+y_{2}\right) \\] and scalar multiplication defined by \\[ \alpha \circ\left(x_{1}, x_{2}\right)=\left(\alpha x_{1}, x_{2}\right) \\] Scalar multiplication for this system is defined in an unusual way, and consequently we use the symbol o to avoid confusion with the ordinary scalar multiplication of row vectors. Is \(V\) a vector space with these operations? Justify your answer.

Determine whether the following are spanning sets for \(\mathbb{R}^{2}:\) (a) \(\left\\{\left(\begin{array}{l}2 \\\ 1\end{array}\right),\left(\begin{array}{l}3 \\ 2\end{array}\right)\right\\}\) (b) \(\left\\{\left(\begin{array}{l}2 \\\ 3\end{array}\right),\left(\begin{array}{l}4 \\ 6\end{array}\right)\right\\}\) (c) \(\left\\{\left(\begin{array}{r}-2 \\\ 1\end{array}\right),\left(\begin{array}{l}1 \\\ 3\end{array}\right),\left(\begin{array}{l}2 \\ 4\end{array}\right)\right\\}\) (d) \(\left\\{\left(\begin{array}{r}-1 \\\ 2\end{array}\right),\left(\begin{array}{r}1 \\\ -2\end{array}\right),\left(\begin{array}{r}2 \\\ -4\end{array}\right)\right\\}\) (e) \(\left\\{\left(\begin{array}{l}1 \\\ 2\end{array}\right),\left(\begin{array}{r}-1 \\ 1\end{array}\right)\right\\}\)

Let \(A\) be an \(m \times n\) matrix. Prove that \\[ \operatorname{rank}(A) \leq \min (m, n) \\]

Which of the sets that follow are spanning sets for \(\mathbb{R}^{3} ?\) Justify your answers. (a) \(\left\\{(1,0,0)^{T},(0,1,1)^{T},(1,0,1)^{T}\right\\}\) (b) \(\left\\{(1,0,0)^{T},(0,1,1)^{T},(1,0,1)^{T},(1,2,3)^{T}\right\\}\) (c) \(\left\\{(2,1,-2)^{T},(3,2,-2)^{T},(2,2,0)^{T}\right\\}\) (d) \(\left\\{(2,1,-2)^{T},(-2,-1,2)^{T},(4,2,-4)^{T}\right\\}\) (e) \(\left\\{(1,1,3)^{T},(0,2,1)^{T}\right\\}\)

Let \(R^{+}\) denote the set of positive real numbers. Define the operation of scalar multiplication, denoted \(\circ,\) by \\[ \alpha \circ x=x^{\alpha} \\] for each \(x \in R^{+}\) and for any real number \(\alpha\). Define the operation of addition, denoted \(\oplus, \mathrm{by}\) \\[ x \oplus y=x \cdot y \quad \text { for all } \quad x, y \in R^{+} \\] Thus, for this system, the scalar product of -3 times \(\frac{1}{2}\) is given by \\[ -3 \circ \frac{1}{2}=\left(\frac{1}{2}\right)^{-3}=8 \\] and the sum of 2 and 5 is given by \\[ 2 \oplus 5=2 \cdot 5=10 \\] Is \(R^{+}\) a vector space with these operations? Prove your answer.

Let \(A\) and \(B\) be row equivalent matrices. (a) Show that the dimension of the column space of \(A\) equals the dimension of the column space of \(B\) (b) Are the column spaces of the two matrices necessarily the same? Justify your answer.

Let \(R\) denote the set of real numbers. Define scalar multiplication by \(\alpha x=\alpha \cdot x \quad\) (the usual multiplication of real numbers) and define addition, denoted \(\oplus,\) by \(x \oplus y=\max (x, y) \quad\) (the maximum of the two numbers Is \(R\) a vector space with these operations? Prove your answer.

Prove that any finite set of vectors that contains the zero vector must be linearly dependent.

Let \(A\) be a \(4 \times 3\) matrix and let \(\mathbf{b} \in \mathbb{R}^{4}\). How many possible solutions could the system \(A \mathbf{x}=\mathbf{b}\) have if \(N(A)=\\{0\\} ?\) Answer the same question in the case \(N(A) \neq\\{0\\} .\) Explain your answers.