Define the linear transformation $T: R^{n} \rightarrow R^{m}$ by T(v) = Av. Find the dimensions of $R^n$ and $R^m$.

$$A=\left[\begin{array}{rrrrr} {-1} & {2} & {1} & {3} & {4} \\ {0} & {0} & {2} & {-1} & {0} \end{array}\right]$$

Let $T: R^{4} \rightarrow R^{3}$ be the linear transformation represented by T(x) = Ax, where

$$A=\left[\begin{array}{rrrr} {1} & {-2} & {1} & {0} \\ {0} & {1} & {2} & {3} \\ {0} & {0} & {0} & {1} \end{array}\right]$$

(a) Find the dimension of the domain. (b) Find the dimension of the range. (c) Find the dimension of the kernel. (d) Is T one-to-one? Explain. (e) Is T onto? Explain. (f) ls T an isomorphism? Explain.

In each case, assume that T is a linear transformation. (a) If

$$T : V \rightarrow \mathbb { R }$$

and

$$T \left( \mathbf { v } _ { 1 } \right) = 1 , T \left( \mathbf { v } _ { 2 } \right) = - 1$$

, find

$$T \left( 3 \mathbf { v } _ { 1 } - 5 \mathbf { v } _ { 2 } \right)$$

. (b) If

$$T : V \rightarrow \mathbb { R }$$

and

$$T \left( \mathbf { v } _ { 1 } \right) = 2 , T \left( \mathbf { v } _ { 2 } \right) = - 3$$

, find

$$T \left( 3 \mathbf { v } _ { 1 } + 2 \mathbf { v } _ { 2 } \right)$$

. (c) If

$$T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$$

and

$$T \left[ \begin{array} { l } { 1 } \\ { 3 } \end{array} \right] = \left[ \begin{array} { l } { 1 } \\ { 1 } \end{array} \right] , T \left[ \begin{array} { l } { 1 } \\ { 1 } \end{array} \right] = \left[ \begin{array} { l } { 0 } \\ { 1 } \end{array} \right]$$

, and

$$T \left[ \begin{array} { r } { - 1 } \\ { 3 } \end{array} \right]$$

. (d) If

$$T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$$

and

$$T \left[ \begin{array} { l } { 1 } \\ { - 1 } \end{array} \right] = \left[ \begin{array} { l } { 0 } \\ { 1 } \end{array} \right] , T \left[ \begin{array} { l } { 1 } \\ { 1 } \end{array} \right] = \left[ \begin{array} { l } { 1 } \\ { 0 } \end{array} \right]$$

, find

$$T \left[ \begin{array} { r } { 1 } \\ { - 7 } \end{array} \right]$$

. (e) If

$$T : \mathbf { P } _ { 2 } \rightarrow \mathbf { P } _ { 2 }$$

and T(x+1)=x,

$$T ( x - 1 ) = 1 , T \left( x ^ { 2 } \right) = 0$$

, find T(2+3x-

$$x^2$$

). (f) If

$$T : \mathbf { P } _ { 2 } \rightarrow \mathbb { R }$$

and T(x+2)=1, T(1)=5,

$$T \left( x ^ { 2 } + x \right) = 0$$

, find T(2-x+

$$3x^2$$

).

Are our concepts limited in principle by our everyday experiences or is this only our conceptual starting point? How is this question related to a resolution of the wave-particle duality?

In each of the following cases, find all possible products

$$A^2$$

, AB, AC, and so on. (a)

$$A = \left[ \begin{array} { r r r } { 1 } & { 2 } & { 3 } \\ { - 1 } & { 0 } & { 0 } \end{array} \right] , B = \left[ \begin{array} { r r } { 1 } & { - 2 } \\ { \frac { 1 } { 2 } } & { 3 } \end{array} \right] , C = \left[ \begin{array} { r r } { - 1 } & { 0 } \\ { 2 } & { 5 } \\ { 0 } & { 5 } \end{array} \right]$$

, (b)

$$A = \left[ \begin{array} { r r r } { 1 } & { 2 } & { 4 } \\ { 0 } & { 1 } & { - 1 } \end{array} \right] , B = \left[ \begin{array} { r r } { - 1 } & { 6 } \\ { 1 } & { 0 } \end{array} \right] , C = \left[ \begin{array} { r r } { 2 } & { 0 } \\ { - 1 } & { 1 } \\ { 1 } & { 2 } \end{array} \right]$$

Let F be a fixed 3 x 2 matrix, and let H be the set of all matrices A in M$_{2 \times 4}$ with the property that FA = 0 (the zero matrix in M$_{3 \times 4})$. Determine if H is a subspace of M$_{2 \times 4}$.

An oil tanker is leaking oil at the rate given (in barrels per hour) by $L^{\prime}(t)=\frac{80 \ln (t+1)}{t+1}$, where t is the time (in hours) after the tanker hits a hidden rock (when t=0). a. Find the total number of barrels that the ship will leak on the first day. b. Find the total number of barrels that the ship will leak on the second day. c. What is happening over the long run to the amount of oil leaked per day?

Determine if the set H of all matrices of the form

$$\left[\begin{array}{ll}a & b \\ 0 & d\end{array}\right]$$

is a subspace of M$_{2 \times 2}$.

Give an example to show that the product of symmetric matrices is not necessarily symmetric.

Evaluate the integral. $\int \frac{\csc ^{2} x}{1+\cot x} d x$

Examine $f(w, z)=w^{3}+z^{3}-3 w z+5$ for relative extrema.

Express the quantity as a single logarithm. $\ln (a+b)+\ln (a-b)-2 \ln c$

Explain how implicit differentiation works.

What were Keynes's three conjectures about the consumption function?